expressed by Serieses of periodic Terms. 163 



differentials; and it is further proved that the characleilstic 

 property of the terms of the series is independent of the ex- 

 pression /(fl, \{>), being derived entirely from the functions 

 P,- produced by the expansion of the radical. The plain in- 

 ference seems to be, that it will be impossible to deduce ge- 

 nerally the value of y"(fl, 4/) from a series the distinguishing 

 character of which is independent of that function: yet this is 

 what M. Poisson undertakes to accomplish by an artifice of 

 calculation. 



The formulas of M. Poisson are these*, viz. 



^ _ _1 ^'T .2^ ( l-a=') / (fl',4>') sin9'<ie'<JCT^ . 



4. yoyo (i~2«;;.f«^)t ' I (1.) 



JJ — cos 9 cos fl' — sin fl sin fl' cos (vf/ — \^'),J 

 a being less than 1, but approaching to it indefinitely: and 

 from this it is proposed to deduce, in the most general manner, 

 the equation 



X' =/(9, ^\ (2.) 



X' representing the value of the double integral X when 

 a = 1. 



If we suppose that_/(5', vj/') is constant in the small extent 

 for which the increments of X are sensible, the first of the 

 equations (1.) may be thus written: 



^_'^^ ^'^^o^o (l-2«p + a^)i ' 



which will coincide with the equation (2.), because the value 

 of the integral is 1 when « = 1, as it is easy to prove in many 

 ways. Now it is presumed that y(6', \J/') maybe supposed 

 constant, because the numerator of the differential is always 

 small on account of the small multiplier (1 — a-), while the de- 

 nominator increases rapidly, and quickly becomes so consi- 

 derable as to make the increments of the integral insensible. 



Such is the demonstration of M. Poisson as it appeared in 

 1823 in the Journal de I'Ecole Polytechnique. In the Conn, des 

 Terns for 1 829, and in the Theory of Heat the author, in order 

 to make the matter plainer, supposes that the arcs fl' and ■!/ 

 vary from the initial values 9 and ■i^ to 9+j/ and \I'+;:, y and z 

 being small increments: in consecjuence 

 ./(«',>') =/(fl+j/, ^1/ + ^) =/(«,vl') + ?: and by substitu- 

 ting this value in the first of the equations (I.), we obtain 

 X'=/(9,4')+X", 



X 



"=/T 



z ( 1 -«^)^sina'J6^/^{/' 

 (1— 2a/j + «-)« 



• Thioiie de la Ckalcur, p. 2] 3. 

 S2 



