226 Notices respecting New Books. 



considerably when the properties of the third class of integrals are 

 considered, and those of the 9 functions (in terms of which the 

 elliptic functions can be expressed) as well as the general theory of 

 the transformation of the elliptic functions, of which Landen's 

 theorem is in fact a particular case. 



Professor Cayley has taken a great deal of pains to clear up the 

 difficulties to be met with at the entrance of the subject. But 

 when all has been done, the elliptic functions will probably appear 

 to most students at first sight somewhat intangible entities. It is 

 perhaps with a view to this circumstance that the following para- 

 graph was written, which we venture to extract as an admirable 

 specimen of elementary exposition: — "In further illustration" of 

 the elliptic functions sn u, cnw, &nu, "suppose that the theory of 

 the circular functions sine, cosine was unknown, and that we had de- 

 fined Yx to be the function 



f dx 



J V(1-* 2 )' 



Then taking the variables x and y to be connected by the differen- 

 tial equation 



dx i _ d V_ =0 

 V(1-*V V(W) ' l 

 and supposing that z is the value of y answering to a? = 0, we have 



Yx + Fy=¥z. 

 But the differential equation admits of algebraic integration : and 

 determining in each case the constant by the condition that for 

 x=0, y shall be — z, the algebraic iutegral maybe expressed in two 



«y- v(i-* 2 )V(i-2/ 2 )= vXi-z" 2 ); 



so that either of these equations represents the above-mentioned 

 transcendental integral; and thus we have a circular theory pre- 

 cisely analogous to the elliptic theory in its original form. But 

 here the function Fa; is the inverse function sin- 1 x, and the last- 

 mentioned two equations are the equivalents of the equation 



sin -1 x + sin -1 y= sin -1 z, 

 w T hence writing sm.— 1 oj==u, sin -1 y=v, and therefore x=s'ma, 

 y = sin v, z = sin (u + v) ; and also assuming sj ( 1 — sin 2 u) = cos u, and 

 therefore V(l— sin 2 v)=cosv,and \((l — sm 2 (u+i;)) = cos(w-fv), 

 the equations in question become 



sin(it-f-t')= sin u cos v + sin v cos u, 



cos (u -\- v) = cos u cos v — sin u sin v, 

 and it is clearly convenient to use these functions sin, ccs in 

 place of F, denoting as above sin -1 The circular theory 



gives rise to a numerical transcendent 7r, viz. -^ = — — is 



such a quantity that sin [- = 1, cos- =0, - being the smallest po- 



sitive value of the argument for which the two functions have these 



C dx 



values ; and in developing the theory from the integral 1 — , 



J V(l-x~) 



