151 



To make use of these properties, I observe that if we put 



r.to* = «% a = ^.<% b = )).«, (b(9)) 



and substitute for «W, tcO their expressions (Z'"'), (A(5') the series (T'") will take the form 

 (EW) ^ = a + («i,o + Mi.i. z). tf + (m2,o + «s,i. « + wa>«'). <« + ... + utjti.^'.S' + &c. 



1=1+ (tUl,0+W>l.l.»).«+(W2,0 + tt'S,l'2+tt'2,2-2*).«' + ... + W/^'.2".«'+ &C. 



equations which give by differentiation 



d^TaP^ CO'. \i>y^ut,t, + [<+!]«. [<•]".«/+!.... «+ [*]'.[<' + !]". «M-+i. 2 + &c. 



-ijj^ = CO'- en", wv + [<+!]«. [f]". t«,+i,,,. + w-c^'+i]". %.,/+,. 2 + &c. 



and therefore, when « = 0, z = 0, 



in order therefore to obtain general expressions for the constants utf„ i,t,t, it is sufficient to 

 calculate expressions for these partial differentials of ^, „. Now, if from the two equations (S'"), 

 [68], we subtract the two others 



X' = 



dx' d*x' , dW 



which result from the formulae (U'") of the same paragraph; if we then eKminate 5» —mV, 

 and put for abridgment, 



^1^ */■„ l.^^t X '^''=' -2 , d^x . . . rf=x' . . . dx' 



da' . " T- "V. - -^ 



^ . Ha, ^) + *• ^ • «'+ S-* + - S' • -'+*-=-^ • ^(- ^) = 

 we shall have the two following equations, 



a=s:ur + F, b^ = tt>V + 2* ; (GW) 



which, when we put 



« = ««;% u>V = «*, azr^i*, bssftf, 

 become 



C^^+ZlCi, 0, «* = 1 + 2^{C 1, 0; (HW) 



^ ?i, being functions such that 



VOL. XV. V 



