134 



... dec, 1 da ^ dfi ^ dfi 1 



in which, at present, —7- = , -jy- = 0, -7- =0, -jr- = , 



'^ da ^^ do da do j, 



cP» d'V d'» d^V dU d'V 



'+*L^r.-+^IS-»^+^-'i + - 



da'' da? ' da.db dx\dy' db'' dxJf 



d^li _ d'V d^p. _ d^V d^_fV_ 

 da'' ~dx''.dy' daM~ dx.dy^ ' db^ ~ dj^' 



And if wc substitute these expressions for »„ ^„ in the two series for a;' and y, we shall get two 

 other series of the form 



C , dx> ^ dx' , , ^ Cd^x' ^ , „ d'x' . , d^af ,A , . 



dx' i dj^ ^ di/ ^ dj/ ^ 



in which, at present, ^ = -- . rfa" = «' 17 = ^' -rfr= ^' 



£V__ d'V rfV _ rf^F </V _ d^F 

 rfa* "*'■ rfi' ' da.db~^^' dx'dy' db^ ~^" rfx.rf/ ' 



^Y__ rf^F _^Y__ <^'F rfy _ ^'F 



rfa' ~^^' dx'^.dy' lkUb~^'" dx.df ' db* ~^'" Ibf ' 



and in which the other coefficients can also be calculated by means of the characteristic function. 

 This being laid down, let us put y = r. cos. v, y == r. sin. w, and let us develope (a) and (6), 

 according to the powers of (r). The developments will be of the form 



■^ 6 = r". to + r'^. tp' + r"".tt)" + 



»!, JK, m"...n, «', n",.. being positive and increasing exponents, which may or may not be 

 fractional, and u, u', u".,m, w', id". ..being functions of the angle (v): which functions, as 

 well as the exponents of the terms that multiply them, we have now to determine. Substituting 

 therefore the values (T"') in the series (S'") we find the following equations; 



dx' 



1st = — r. COS. t) + -7—. (r'"M+...) 



da 



