First composition derivatives from SRK

$$\frac{\partial{} \alpha^r}{{}{\partial \delta}} = \frac{1}{R T_{r}} \left(- R T_{r}\frac{b}{b\rho_r\delta - 1} - \rho_{r} \tau\frac{ a{\left (\tau \right )}}{b \delta \rho_{r} + 1}\right)$$

$$\frac{\partial}{\partial x_i}\left(\frac{\partial{} \alpha^r}{{}{\partial \delta}}\right) = \frac{1}{R T_{r}} \left(-RT_r \frac{(b\delta\rho_r-1)\frac{db}{dx_i}-b\rho_r\delta\frac{db}{dx_i}}{(b \rho_{r}\delta - 1)^2} - \rho_{r} \tau \frac{(b \delta \rho_{r} + 1)\frac{\partial a}{\partial x_i}-a\delta\rho_r\frac{\partial b}{\partial x_i}}{(b \delta \rho_{r} + 1)^2}\right)$$

$$\frac{\partial}{\partial x_i}\left(\frac{\partial{} \alpha^r}{{}{\partial \delta}}\right) = \frac{1}{R T_{r}} \left( \frac{db}{dx_i}\frac{RT_r}{(b \rho_{r}\delta - 1)^2} - \rho_{r} \tau \frac{(b \delta \rho_{r} + 1)\frac{\partial a}{\partial x_i}-a\delta\rho_r\frac{\partial b}{\partial x_i}}{(b \delta \rho_{r} + 1)^2}\right)$$



In [ ]:

$$\frac{\partial{} \alpha^r}{{\partial \tau}{}} = \frac{1}{R T_{r}} \left(\left(- \tau\frac{\frac{d}{d \tau} a{\left (\tau \right )}}{b} - \frac{a{\left (\tau \right )}}{b} \right) \log{\left (b \delta \rho_{r} + 1 \right )}\right)$$

$$\frac{\partial}{\partial x_i}\left(\frac{\partial{} \alpha^r}{{\partial \tau}{}}\right) = \frac{1}{R T_{r}} \left(\left(- \tau \frac{b\frac{d^2a}{\partial\tau\partial x_i} -\frac{\partial a}{\partial \tau}\frac{\partial b}{\partial x_i}}{b^2} - \frac{b\frac{\partial a}{\partial x_i}-a\frac{\partial b}{\partial x_i}}{b^2} \right) \log{\left (b \delta \rho_{r} + 1 \right )} + \left(- \tau\frac{\frac{d}{d \tau} a{\left (\tau \right )}}{b} - \frac{a{\left (\tau \right )}}{b} \right) \frac{\delta\rho_r}{\left (b \delta \rho_{r} + 1 \right )}\frac{\partial b}{\partial x_i} \right)$$



In [ ]:

$$\frac{\partial{^{2}} \alpha^r}{{}{\partial \delta^{2}}} = \frac{1}{R T_{r}} \left(R T_{r}\frac{1}{\left(\delta - \frac{1}{b \rho_{r}}\right)^{2}} + \rho_{r}^{2} \tau \frac{b a{\left (\tau \right )}}{\left(b \delta \rho_{r} + 1\right)^{2}}\right)$$

$$\frac{\partial}{\partial x_i} \left( \frac{\partial{^{2}} \alpha^r}{{}{\partial \delta^{2}}} \right) = \frac{1}{R T_{r}} \left(R T_{r}\frac{-2\left(\delta - \frac{1}{b \rho_{r}}\right)\left(\frac{1}{b\rho_r^2}\right)}{\left(\delta - \frac{1}{b \rho_{r}}\right)^{4}} + \rho_{r}^{2} \tau \frac{\left(b \delta \rho_{r} + 1\right)^{2}(b\frac{\partial a}{\partial x_i}-a\frac{\partial b}{\partial x_i})-2ba\left(b \delta \rho_{r} + 1\right)\delta\rho_r\frac{\partial b}{\partial x_i}}{\left(b \delta \rho_{r} + 1\right)^{2}}\right) $$

$$\frac{\partial}{\partial x_i} \left( \frac{\partial{^{2}} \alpha^r}{{}{\partial \delta^{2}}} \right) = \frac{1}{R T_{r}} \left(-\frac{2R T_{r}}{b^2\rho_r\left(\delta - \frac{1}{b \rho_{r}}\right)^{3}} + \rho_{r}^{2} \tau\left( \left(b\frac{\partial a}{\partial x_i}-a\frac{\partial b}{\partial x_i}\right) -\frac{2ba\delta\rho_r\frac{\partial b}{\partial x_i}}{\left(b \delta \rho_{r} + 1\right)}\right)\right) $$



In [ ]:

$$\frac{\partial{^{2}} \alpha^r}{{\partial \tau}{\partial \delta}} = -\frac{\rho_r}{R T_{r} } \left(\frac{\tau \frac{da}{d \tau}+a}{b \delta \rho_{r} + 1} \right)$$

$$\frac{\partial}{\partial x_i}\left(\frac{\partial{^{2}} \alpha^r}{{\partial \tau}{\partial \delta}}\right) = -\frac{\rho_r}{R T_{r} } \left( \frac{(b \delta \rho_{r} + 1)(\tau \frac{\partial^2a}{\partial \tau \partial x_i}+\frac{\partial a}{\partial x_i})-(\tau \frac{da}{d \tau}+a)\delta\rho_r\frac{\partial b}{\partial x_i}}{(b \delta \rho_{r} + 1)^2} \right)$$



In [ ]:

$$\frac{\partial{^{2}} \alpha^r}{{\partial \tau^{2}}{}} = - \frac{1}{R T_{r} } \left(\tau \frac{d^{2}}{d \tau^{2}} a{\left (\tau \right )} + 2 \frac{d}{d \tau} a{\left (\tau \right )}\right) \frac{\log{\left (b \delta \rho_{r} + 1 \right )}}{b}$$

$$\frac{\partial}{\partial x_i}\left(\frac{\partial{^{2}} \alpha^r}{{\partial \tau^{2}}{}}\right) = - \frac{1}{R T_{r} } \left[\left(\tau \frac{\partial^{3}a}{\partial \tau^{2}\partial x_i} + 2 \frac{\partial^2a}{\partial \tau\partial x_i} \right) \frac{\log{\left (b \delta \rho_{r} + 1 \right )}}{b} + \left(\tau \frac{d^{2}}{d \tau^{2}} a{\left (\tau \right )} + 2 \frac{d}{d \tau} a{\left (\tau \right )}\right) \frac{\partial b}{\partial x_i}\frac{\frac{b\delta\rho_r}{\left (b \delta \rho_{r} + 1 \right )} - \log{\left (b \delta \rho_{r} + 1 \right )}}{b^2} \right]$$



In [ ]:

$$\frac{\partial{^{3}} \alpha^r}{{}{\partial \delta^{3}}} = - \frac{1}{R T_{r}} \left(2 R T_{r}\frac{1}{\left(\delta - \frac{1}{b \rho_{r}}\right)^{3}} + 2 \rho_{r}^{3} \tau \frac{b^{2} a{\left (\tau \right )}}{\left(b \delta \rho_{r} + 1\right)^{3}}\right)$$

$$\frac{\partial}{\partial x_i}\left(\frac{\partial{^{3}} \alpha^r}{{}{\partial \delta^{3}}}\right) = - \frac{1}{R T_{r}} \left(2 R T_{r}\frac{-3\frac{1}{b^2\rho_r}\frac{\partial b}{\partial x_i}}{\left(\delta - \frac{1}{b \rho_{r}}\right)^{4}} + 2 \rho_{r}^{3} \tau \frac{\left(b \delta \rho_{r} + 1\right)^{3}(2ba\frac{\partial b}{\partial x_i}+b^2\frac{\partial a}{\partial x_i})-(b^2a)3\left(b \delta \rho_{r} + 1\right)^{2}\delta\rho_r\frac{\partial b}{\partial x_i}}{\left(b \delta \rho_{r} + 1\right)^{6}}\right)$$

$$\frac{\partial}{\partial x_i}\left(\frac{\partial{^{3}} \alpha^r}{{}{\partial \delta^{3}}}\right) = - \frac{1}{R T_{r}} \left(\frac{-6 R T_{r}}{b^2\rho_r}\frac{\frac{\partial b}{\partial x_i}}{\left(\delta - \frac{1}{b \rho_{r}}\right)^{4}} + 2 \rho_{r}^{3} \tau \frac{\left(b \delta \rho_{r} + 1\right)(2ba\frac{\partial b}{\partial x_i}+b^2\frac{\partial a}{\partial x_i})-3b^2a\delta\rho_r\frac{\partial b}{\partial x_i}}{\left(b \delta \rho_{r} + 1\right)^{4}}\right)$$



In [ ]:

$$\frac{\partial{^{3}} \alpha^r}{{\partial \tau}{\partial \delta^{2}}} = \frac{\rho_{r}^{2}}{R T_{r} }\frac{b \left(\tau \frac{d}{d \tau} a{\left (\tau \right )} + a{\left (\tau \right )}\right)}{\left(b \delta \rho_{r} + 1\right)^{2}}$$

$$\frac{\partial}{\partial x_i}\left(\frac{\partial{^{3}} \alpha^r}{{\partial \tau}{\partial \delta^{2}}}\right) = \frac{\rho_{r}^{2}}{R T_{r} } \frac{\left(b \delta \rho_{r} + 1\right)^2\left[\frac{\partial b}{\partial x_i}\left(\tau \frac{da}{d \tau} + a\right) + b\left(\tau \frac{d^2a}{\partial \tau\partial x_i} + \frac{\partial a}{\partial x_i}\right)\right] -2b\delta \rho_{r} \left(b \delta \rho_{r} + 1\right) \left(\tau \frac{da}{d \tau} + a\right) \frac{\partial b}{\partial x_i}}{\left(b \delta \rho_{r} + 1\right)^{4}}$$



In [ ]:

$$\frac{\partial{^{3}} \alpha^r}{{\partial \tau^{2}}{\partial \delta}} = - \frac{\rho_{r}}{R T_{r}}\frac{1}{ \left(b \delta \rho_{r} + 1\right)} \left(\tau \frac{d^{2}}{d \tau^{2}} a{\left (\tau \right )} + 2 \frac{d}{d \tau} a{\left (\tau \right )}\right)$$

$$\frac{\partial}{\partial x_i}\left(\frac{\partial{^{3}} \alpha^r}{{\partial \tau^{2}}{\partial \delta}}\right) = - \frac{\rho_{r}}{R T_{r}}\left[\frac{1}{ \left(b \delta \rho_{r} + 1\right)} \left(\tau \frac{d^{3}a}{d \tau^{2}\partial x_i} + 2 \frac{\partial^2a}{\partial\tau\partial x_i} \right) + \frac{-\delta \rho_{r}}{ \left(b \delta \rho_{r} + 1\right)^2}\frac{\partial b}{\partial x_i} \left(\tau \frac{d^{2}}{d \tau^{2}} a{\left (\tau \right )} + 2 \frac{d}{d \tau} a{\left (\tau \right )}\right)\right]$$



In [ ]:

$$\frac{\partial{^{3}} \alpha^r}{{\partial \tau^{3}}{}} = - \frac{1}{R T_{r}} \left(\tau \frac{d^{3}}{d \tau^{3}} a{\left (\tau \right )} + 3 \frac{d^{2}}{d \tau^{2}} a{\left (\tau \right )}\right) \frac{\log{\left (b \delta \rho_{r} + 1 \right )}}{b} $$

$$\frac{\partial}{\partial x_i}\left(\frac{\partial{^{3}} \alpha^r}{{\partial \tau^{3}}{}}\right) = - \frac{1}{R T_{r} } \left[\left(\tau \frac{\partial^{4}a}{\partial \tau^{3}\partial x_i} + 3 \frac{\partial^3a}{\partial \tau^2\partial x_i} \right) \frac{\log{\left (b \delta \rho_{r} + 1 \right )}}{b} + \left(\tau \frac{d^{3}}{d \tau^{3}} a{\left (\tau \right )} + 3 \frac{d^2}{d \tau^2} a{\left (\tau \right )}\right) \frac{\partial b}{\partial x_i}\frac{\frac{b\delta\rho_r}{\left (b \delta \rho_{r} + 1 \right )} - \log{\left (b \delta \rho_{r} + 1 \right )}}{b^2} \right]$$



In [ ]:

$$\frac{\partial{^{4}} \alpha^r}{{}{\partial \delta^{4}}} = - \frac{1}{R T_{r}} \left(6 R T_{r}\frac{1}{\left(\delta - \frac{1}{b \rho_{r}}\right)^{4}} + 6 \rho_{r}^{4} \tau \frac{b^{3} a{\left (\tau \right )}}{\left(b \delta \rho_{r} + 1\right)^{4}}\right)$$

$$\frac{\partial}{\partial x_i}\left(\frac{\partial{^{4}} \alpha^r}{{}{\partial \delta^{4}}}\right) = - \frac{1}{R T_{r}} \left(6 R T_{r}\frac{-4\frac{1}{b^2\rho_r}\frac{\partial b}{\partial x_i}}{\left(\delta - \frac{1}{b \rho_{r}}\right)^{5}} + 6 \rho_{r}^{4} \tau \frac{\left(b \delta \rho_{r} + 1\right)^{4}(3b^2a\frac{\partial b}{\partial x_i}+b^3\frac{\partial a}{\partial x_i})-(b^3a)4\left(b \delta \rho_{r} + 1\right)^{3}\delta\rho_r\frac{\partial b}{\partial x_i}}{\left(b \delta \rho_{r} + 1\right)^{8}}\right)$$



In [ ]:

$$\frac{\partial{^{4}} \alpha^r}{{\partial \tau}{\partial \delta^{3}}} = - \frac{2 \rho_{r}^{3}}{R T_{r} }\frac{ b^{2}}{\left(b \delta \rho_{r} + 1\right)^{3}} \left(\tau \frac{d}{d \tau} a{\left (\tau \right )} + a{\left (\tau \right )}\right)$$

$$\frac{\partial}{\partial x_i}\left(\frac{\partial{^{4}} \alpha^r}{{\partial \tau}{\partial \delta^{3}}}\right) = - \frac{2 \rho_{r}^{3}}{R T_{r} }\left[\frac{b^{2}}{\left(b \delta \rho_{r} + 1\right)^{3}} \left(\tau \frac{\partial^2a}{\partial\tau\partial x_i} + \frac{\partial a}{\partial x_i}\right) + \frac{\left(b \delta \rho_{r} + 1\right)^{3}2b\frac{db}{dx_i}-b^23\left(b \delta \rho_{r} + 1\right)^2\delta\rho_r\frac{db}{dx_i}}{\left(b \delta \rho_{r} + 1\right)^{6}} \left(\tau \frac{d}{d \tau} a{\left (\tau \right )} + a{\left (\tau \right )}\right)\right]$$

$$\frac{\partial}{\partial x_i}\left(\frac{\partial{^{4}} \alpha^r}{{\partial \tau}{\partial \delta^{3}}}\right) = - \frac{2 \rho_{r}^{3}}{R T_{r} }\left[\frac{b^{2}}{\left(b \delta \rho_{r} + 1\right)^{3}} \left(\tau \frac{\partial^2a}{\partial\tau\partial x_i} + \frac{\partial a}{\partial x_i}\right) + b\frac{\partial b}{\partial x_i}\frac{2\left(\delta \rho_{r} + 1\right)-3b\delta\rho_r}{\left(b \delta \rho_{r} + 1\right)^{4}} \left(\tau \frac{d}{d \tau} a{\left (\tau \right )} + a{\left (\tau \right )}\right)\right]$$



In [ ]:

$$\frac{\partial{^{4}} \alpha^r}{{\partial \tau^{2}}{\partial \delta^{2}}} = \frac{\rho_{r}^{2}}{R T_{r}}\frac{b}{ \left(b \delta \rho_{r} + 1\right)^{2}} \left(\tau \frac{d^{2}}{d \tau^{2}} a{\left (\tau \right )} + 2 \frac{d}{d \tau} a{\left (\tau \right )}\right)$$

$$\frac{\partial}{\partial x_i}\left(\frac{\partial{^{4}} \alpha^r}{{\partial \tau^{2}}{\partial \delta^{2}}}\right) = \frac{\rho_{r}^{2}}{R T_{r}}\left[\frac{b}{\left(b \delta \rho_{r} + 1\right)^{2}} \left(\tau \frac{\partial^{3}a}{\partial \tau^{2}\partial x_i} + 2 \frac{d^2a}{d \tau\partial x_i}\right) + \frac{\left(b \delta \rho_{r} + 1\right)^{2}\frac{\partial b}{\partial x_i} -b2\left(b \delta \rho_{r} + 1\right)\delta\rho_r\frac{\partial b}{\partial x_i}}{ \left(b \delta \rho_{r} + 1\right)^{4}} \left(\tau \frac{d^{2}}{d \tau^{2}} a{\left (\tau \right )} + 2 \frac{d}{d \tau} a{\left (\tau \right )}\right)\right]$$

$$\frac{\partial}{\partial x_i}\left(\frac{\partial{^{4}} \alpha^r}{{\partial \tau^{2}}{\partial \delta^{2}}}\right) = \frac{\rho_{r}^{2}}{R T_{r}}\left[\frac{b}{\left(b \delta \rho_{r} + 1\right)^{2}} \left(\tau \frac{\partial^{3}a}{\partial \tau^{2}\partial x_i} + 2 \frac{d^2a}{d \tau\partial x_i}\right) + \frac{\left(b \delta \rho_{r} + 1\right)\frac{\partial b}{\partial x_i} -2b\delta\rho_r\frac{\partial b}{\partial x_i}}{ \left(b \delta \rho_{r} + 1\right)} \left(\tau \frac{d^{2}}{d \tau^{2}} a{\left (\tau \right )} + 2 \frac{d}{d \tau} a{\left (\tau \right )}\right)\right]$$



In [ ]:

$$\frac{\partial{^{4}} \alpha^r}{{\partial \tau^{3}}{\partial \delta}} = - \frac{\rho_{r}}{R T_{r}}\frac{\tau \frac{d^{3}a}{d \tau^{3}} + 3 \frac{d^{2}a}{d \tau^{2}}}{ \left(b \delta \rho_{r} + 1\right)}$$

$$\frac{\partial}{\partial x_i}\left(\frac{\partial{^{4}} \alpha^r}{{\partial \tau^{3}}{\partial \delta}}\right) = - \frac{\rho_{r}}{R T_{r}}\frac{\left(b \delta \rho_{r} + 1\right)\left(\tau \frac{d^{4}a}{d \tau^{3}\partial x_i} + 3 \frac{\partial^{3}a}{\partial \tau^{2}\partial x_i}\right) - \left(\tau \frac{d^{3}a}{d \tau^{3}} + 3 \frac{d^{2}a}{d \tau^{2}}\right)\delta\rho_r\frac{\partial b}{\partial x_i}}{ \left(b \delta \rho_{r} + 1\right)^2}$$



In [ ]:

$$\frac{\partial{^{4}} \alpha^r}{{\partial \tau^{4}}{}} = - \frac{1}{R T_{r}} \left(\tau \frac{d^{4}}{d \tau^{4}} a{\left (\tau \right )} + 4 \frac{d^{3}}{d \tau^{3}} a{\left (\tau \right )}\right) \frac{\log{\left (b \delta \rho_{r} + 1 \right )}}{b}$$

$$\frac{\partial}{\partial x_i}\left(\frac{\partial{^{4}} \alpha^r}{{\partial \tau^{4}}{}}\right) = - \frac{1}{R T_{r} } \left[\left(\tau \frac{\partial^{5}a}{\partial \tau^{4}\partial x_i} + 4 \frac{\partial^4a}{\partial \tau^3\partial x_i} \right) \frac{\log{\left (b \delta \rho_{r} + 1 \right )}}{b} + \left(\tau \frac{d^{4}}{d \tau^{4}} a{\left (\tau \right )} + 4 \frac{d^3}{d \tau^3} a{\left (\tau \right )}\right) \frac{\partial b}{\partial x_i}\frac{\frac{b\delta\rho_r}{\left (b \delta \rho_{r} + 1 \right )} - \log{\left (b \delta \rho_{r} + 1 \right )}}{b^2} \right]$$

Derivatives of $a$

$$a = \sum_i\sum_jx_ix_j\sqrt{a_ia_j}(1-k_{ij}) $$$$\frac{\partial a}{\partial \tau} = \sum_i\sum_jx_ix_j\frac{1}{2\sqrt{a_ia_j}}\left(a_i\frac{\partial a_j}{\partial \tau}+a_j\frac{\partial a_i}{\partial \tau}\right)(1-k_{ij}) $$$$\frac{\partial^2 a}{\partial \tau\partial x_i} = \sum_jx_j\frac{1}{2\sqrt{a_ia_j}}\left(a_i\frac{\partial a_j}{\partial \tau}+a_j\frac{\partial a_i}{\partial \tau}\right)(1-k_{ij}) $$$$\frac{\partial^3 a}{\partial \tau\partial x_i\partial x_j} = \frac{1}{2\sqrt{a_ia_j}}\left(a_i\frac{\partial a_j}{\partial \tau}+a_j\frac{\partial a_i}{\partial \tau}\right)(1-k_{ij}) $$



In [1]:

    
from __future__ import division
from sympy import *
from IPython.display import display
init_printing()
tau = symbols('tau', positive = True, real = True)

a_i = symbols('a_i', cls = Function)(tau)
a_j = symbols('a_j', cls = Function)(tau)

for idiff in range(0,5):
    display(simplify(diff(sqrt(a_i*a_j),tau,idiff)))









    





$$\sqrt{\operatorname{a_{i}}{\left (\tau \right )} \operatorname{a_{j}}{\left (\tau \right )}}$$






    





$$\frac{\sqrt{\operatorname{a_{i}}{\left (\tau \right )} \operatorname{a_{j}}{\left (\tau \right )}}}{2 \operatorname{a_{i}}{\left (\tau \right )} \operatorname{a_{j}}{\left (\tau \right )}} \left(\operatorname{a_{i}}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )} + \operatorname{a_{j}}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )}\right)$$






    





$$\frac{\sqrt{\operatorname{a_{i}}{\left (\tau \right )} \operatorname{a_{j}}{\left (\tau \right )}}}{4 \operatorname{a_{i}}^{2}{\left (\tau \right )} \operatorname{a_{j}}^{2}{\left (\tau \right )}} \left(2 \operatorname{a_{i}}^{2}{\left (\tau \right )} \operatorname{a_{j}}{\left (\tau \right )} \frac{d^{2}}{d \tau^{2}}  \operatorname{a_{j}}{\left (\tau \right )} - \operatorname{a_{i}}^{2}{\left (\tau \right )} \left(\frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )}\right)^{2} + 2 \operatorname{a_{i}}{\left (\tau \right )} \operatorname{a_{j}}^{2}{\left (\tau \right )} \frac{d^{2}}{d \tau^{2}}  \operatorname{a_{i}}{\left (\tau \right )} + 2 \operatorname{a_{i}}{\left (\tau \right )} \operatorname{a_{j}}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )} - \operatorname{a_{j}}^{2}{\left (\tau \right )} \left(\frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )}\right)^{2}\right)$$






    





$$\frac{\sqrt{\operatorname{a_{i}}{\left (\tau \right )} \operatorname{a_{j}}{\left (\tau \right )}}}{8 \operatorname{a_{i}}^{3}{\left (\tau \right )} \operatorname{a_{j}}^{3}{\left (\tau \right )}} \left(4 \operatorname{a_{i}}^{3}{\left (\tau \right )} \operatorname{a_{j}}^{2}{\left (\tau \right )} \frac{d^{3}}{d \tau^{3}}  \operatorname{a_{j}}{\left (\tau \right )} - 6 \operatorname{a_{i}}^{3}{\left (\tau \right )} \operatorname{a_{j}}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )} \frac{d^{2}}{d \tau^{2}}  \operatorname{a_{j}}{\left (\tau \right )} + 3 \operatorname{a_{i}}^{3}{\left (\tau \right )} \left(\frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )}\right)^{3} + 4 \operatorname{a_{i}}^{2}{\left (\tau \right )} \operatorname{a_{j}}^{3}{\left (\tau \right )} \frac{d^{3}}{d \tau^{3}}  \operatorname{a_{i}}{\left (\tau \right )} + 6 \operatorname{a_{i}}^{2}{\left (\tau \right )} \operatorname{a_{j}}^{2}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )} \frac{d^{2}}{d \tau^{2}}  \operatorname{a_{j}}{\left (\tau \right )} + 6 \operatorname{a_{i}}^{2}{\left (\tau \right )} \operatorname{a_{j}}^{2}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )} \frac{d^{2}}{d \tau^{2}}  \operatorname{a_{i}}{\left (\tau \right )} - 3 \operatorname{a_{i}}^{2}{\left (\tau \right )} \operatorname{a_{j}}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )} \left(\frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )}\right)^{2} - 6 \operatorname{a_{i}}{\left (\tau \right )} \operatorname{a_{j}}^{3}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )} \frac{d^{2}}{d \tau^{2}}  \operatorname{a_{i}}{\left (\tau \right )} - 3 \operatorname{a_{i}}{\left (\tau \right )} \operatorname{a_{j}}^{2}{\left (\tau \right )} \left(\frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )}\right)^{2} \frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )} + 3 \operatorname{a_{j}}^{3}{\left (\tau \right )} \left(\frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )}\right)^{3}\right)$$






    





$$\frac{\sqrt{\operatorname{a_{i}}{\left (\tau \right )} \operatorname{a_{j}}{\left (\tau \right )}}}{16 \operatorname{a_{i}}^{4}{\left (\tau \right )} \operatorname{a_{j}}^{4}{\left (\tau \right )}} \left(8 \operatorname{a_{i}}^{4}{\left (\tau \right )} \operatorname{a_{j}}^{3}{\left (\tau \right )} \frac{d^{4}}{d \tau^{4}}  \operatorname{a_{j}}{\left (\tau \right )} - 16 \operatorname{a_{i}}^{4}{\left (\tau \right )} \operatorname{a_{j}}^{2}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )} \frac{d^{3}}{d \tau^{3}}  \operatorname{a_{j}}{\left (\tau \right )} - 12 \operatorname{a_{i}}^{4}{\left (\tau \right )} \operatorname{a_{j}}^{2}{\left (\tau \right )} \left(\frac{d^{2}}{d \tau^{2}}  \operatorname{a_{j}}{\left (\tau \right )}\right)^{2} + 36 \operatorname{a_{i}}^{4}{\left (\tau \right )} \operatorname{a_{j}}{\left (\tau \right )} \left(\frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )}\right)^{2} \frac{d^{2}}{d \tau^{2}}  \operatorname{a_{j}}{\left (\tau \right )} - 15 \operatorname{a_{i}}^{4}{\left (\tau \right )} \left(\frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )}\right)^{4} + 8 \operatorname{a_{i}}^{3}{\left (\tau \right )} \operatorname{a_{j}}^{4}{\left (\tau \right )} \frac{d^{4}}{d \tau^{4}}  \operatorname{a_{i}}{\left (\tau \right )} + 16 \operatorname{a_{i}}^{3}{\left (\tau \right )} \operatorname{a_{j}}^{3}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )} \frac{d^{3}}{d \tau^{3}}  \operatorname{a_{j}}{\left (\tau \right )} + 16 \operatorname{a_{i}}^{3}{\left (\tau \right )} \operatorname{a_{j}}^{3}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )} \frac{d^{3}}{d \tau^{3}}  \operatorname{a_{i}}{\left (\tau \right )} + 24 \operatorname{a_{i}}^{3}{\left (\tau \right )} \operatorname{a_{j}}^{3}{\left (\tau \right )} \frac{d^{2}}{d \tau^{2}}  \operatorname{a_{i}}{\left (\tau \right )} \frac{d^{2}}{d \tau^{2}}  \operatorname{a_{j}}{\left (\tau \right )} - 24 \operatorname{a_{i}}^{3}{\left (\tau \right )} \operatorname{a_{j}}^{2}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )} \frac{d^{2}}{d \tau^{2}}  \operatorname{a_{j}}{\left (\tau \right )} - 12 \operatorname{a_{i}}^{3}{\left (\tau \right )} \operatorname{a_{j}}^{2}{\left (\tau \right )} \left(\frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )}\right)^{2} \frac{d^{2}}{d \tau^{2}}  \operatorname{a_{i}}{\left (\tau \right )} + 12 \operatorname{a_{i}}^{3}{\left (\tau \right )} \operatorname{a_{j}}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )} \left(\frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )}\right)^{3} - 16 \operatorname{a_{i}}^{2}{\left (\tau \right )} \operatorname{a_{j}}^{4}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )} \frac{d^{3}}{d \tau^{3}}  \operatorname{a_{i}}{\left (\tau \right )} - 12 \operatorname{a_{i}}^{2}{\left (\tau \right )} \operatorname{a_{j}}^{4}{\left (\tau \right )} \left(\frac{d^{2}}{d \tau^{2}}  \operatorname{a_{i}}{\left (\tau \right )}\right)^{2} - 12 \operatorname{a_{i}}^{2}{\left (\tau \right )} \operatorname{a_{j}}^{3}{\left (\tau \right )} \left(\frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )}\right)^{2} \frac{d^{2}}{d \tau^{2}}  \operatorname{a_{j}}{\left (\tau \right )} - 24 \operatorname{a_{i}}^{2}{\left (\tau \right )} \operatorname{a_{j}}^{3}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )} \frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )} \frac{d^{2}}{d \tau^{2}}  \operatorname{a_{i}}{\left (\tau \right )} + 6 \operatorname{a_{i}}^{2}{\left (\tau \right )} \operatorname{a_{j}}^{2}{\left (\tau \right )} \left(\frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )}\right)^{2} \left(\frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )}\right)^{2} + 36 \operatorname{a_{i}}{\left (\tau \right )} \operatorname{a_{j}}^{4}{\left (\tau \right )} \left(\frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )}\right)^{2} \frac{d^{2}}{d \tau^{2}}  \operatorname{a_{i}}{\left (\tau \right )} + 12 \operatorname{a_{i}}{\left (\tau \right )} \operatorname{a_{j}}^{3}{\left (\tau \right )} \left(\frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )}\right)^{3} \frac{d}{d \tau} \operatorname{a_{j}}{\left (\tau \right )} - 15 \operatorname{a_{j}}^{4}{\left (\tau \right )} \left(\frac{d}{d \tau} \operatorname{a_{i}}{\left (\tau \right )}\right)^{4}\right)$$



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