402 MISCELLANEOUS PROPOSITIONS. 



Assume that Jacobi's formula is true for a specific value 

 of n ; differentiate both sides with respect to x : thus 



1.3.5...(2n-l)cosn0 



- 



- 7 n - ~ /I - 



dx sin 



Using this result, and also Jacobi's formula, on the right- 

 hand side of (3), we obtain 



(n + 1) -jrr = (* 1) B 1 . 3 . 5 ... (2n 4- 1) cos w0 sin 

 + (- 1)" 1 . 3 . 5 ... (2n + 1) sin w0 cos 

 = (- 1)" 1. 3. 5 ... (2n + 1) sin (n + 1) 0; 

 1.3.5...(2n + l)s 



, - . i 



therefore -- = (- 1) 





This shews that if Jacobi's formula is true for a specific 

 value of n it is true for that value increased by unity ; and 

 it is obviously true when n=l, and when n = 2 : therefore it 

 is true for any positive integral value of w. 



371. The following proposition is useful in some appli- 

 cations of mathematics to natural philosophy : Having given 

 that if a; varies, it must be such a function of the independent 



ftnf* 



variable t, that -j- = ax, where a is some quantity, not neces- 



CLt 



sarily constant, which is always finite ; and having given 

 that x is zero when t is zero : then it will follow that x 

 cannot vary, or, in other words, that x is always zero. 



Denote x by $ (t). "We know by Art. 101 that 



where 6 is some proper fraction. 



In the present case < (0) = 0, and </>' (dt] = a<f> (6t} t where a 

 is some finite quantity. Thus we have 



(f) = ta$ (8t) t 

 and therefore, if <f> (t) be not zero, 







