12 RESEARCHES ON II. 



and integrating both members, and transposing the terms, we shall have 

 ,-10. r 1 7 r A dd) (? COS. A sin. <& 1 t-, c^ cos. A sin. A 

 (13) J ^3^^= J V^- ih = ¥^-—Wa-^- 



Substituting in (10) the values of the first and third terms as they are given by 

 (13) and (9), we shall have 



f-tj. Csin.^ad^ _ 1^ ,-pr vi jfi\ (^o^- <|> sin. ^ 



^ ^ -' A^ ¥c^ ^ y A ' 



Observation. — The values of the integrals indicated by E and F are the values of 

 the elliptical functions of the first and second kind, the angle ^ being their ampli- 

 tude and c their modules. This being always a fraction, can be expressed by the 

 sine of an angle Q, that is, we may suppose 



c = sin. d. 

 i is also called the complement of the modulus ; therefore we shall have 



h = \/ 1 — c^ = COS. d. 

 As the integrals in ^ are to be taken between the limits (^ = and ^ = Tt, so at 

 their limits U and F will be the double of the complete fwnctions marked E''-, F^. 

 Their value is found in Legendre's Tables, t. 2, p. 288, for every degree of 6. Their 

 logarithms for every tenth part of a degree of the modulus, page 222 et seq. They 

 might be calculated by the formulge 



-*=fC-(i)'«-(y'--(^)"^''-)' , 



whose second members are obtained by developing -r d ^ and Ad ^ hj means of 



the binomial formula, and making <|> = „ after the integration ; but easier formulae 



are given in Legendre's work, which should be consulted in every case.^ 



After these preparations, the integration of formulge (5), §. 2, is easily performed. 

 The value of X, for instance, may be written 



2 7£nR rr2 sin. ^ cos. ^ cos. a sin. a . 2 sin. a sin.'' ^"1 ^ 



The indefinite integral of which by formulae (7), (13), (14), is 



X= 



2JcnR r2 cos. a • /!■„(? cos. &> sin. A\ 



-^^L^^-""-"b^- Ja ) 



2 sin. a ( J^ {E- W F) - ^.££:W)] + const., 



which being reduced to the due limits, all the algebraical terms vanish, and the 

 integral becomes 



* Although several of the preceding transformations are found in Legendre's work, I did not think unfit 

 to develop them here, because that work may not be in the hands of those who have to peruse this paper. 



