Notices respecting New Boohs. 637 



Good instances are provided in Chap. VI, where the integrand 

 is the reciprocal of a + b cosx + c sin x, . . . . ; aud the method is- 



JJSlx J- Js" dec 

 — ~x — VY' 



T 



where X, T are both quadratics; the substitution y= ^ breaks 



up the integral without preparation into two simple standard 

 forms, circular and hyperbolic. This integral may be cited as 

 the degenerate hyperelliptic integral, where the sextic under the 

 radical breaks up into X 2 Y. So too when X and T are both 

 linear, a degenerate elliptic integral is encountered, the cubic 

 under the radical is X 2 Y, having the repeated factor X. 



Once started on the Elliptic Integral in Chap. XI, and the 

 author finds it difficult to pull up ; we are promised a sequel of 

 complete treatment in vol. II. The lemniscate and cassinian are 

 treated as applications and with elegance, but had better have 

 been delayed till later on. The pious adhesion here to the old 

 standard forms of Legendre should give way to the more elastic- 

 treatment of Weierstrass, as not requiring preparation, taking 

 the irrationality in the integrand as reduced to the square root of 

 a cubic ; but disregarding the undimensional abstraction of the p 

 function, to consider the Second Stage functions of Abel and 

 Jacobi, where the cubic is resolved into factors 



X=4. x — x l . x — x 2 . x— x 3 



in § 388, and to discuss the various forms that arise and the 

 double periodicity, as the variable x traverses the regions 

 bounded by the branch points x v x 2 , x 3 . As indicated above, 

 with x x >x 2 >oc s , the degenerate form is investigated, circular or 

 hyperbolic, as the middle root x 2 moves up to x v or down to x z . 



' The author works the circular and hyperbolic functions together 

 in harness, and so maintains an analogy of assistance in the 

 treatment. Although so innovating as to introduce the elliptic 

 functions, and to employ the Jacobi-G-udermann notation of 

 sn, en, dn, and their inverse, he has stopped short of the modern 

 French abbreviations of sh, ch, th, for the hyperbolic functions, 

 analogous to the old sin, cos, tan, of the circular Trigonometry. 

 Bertrand's tangsecthyp strikes us to-day as a very cumbrous 

 form of tanh, or th. 



A picture of the integration of y=x n is provided in fig. 16, 

 p. 112, where the curve PQ then divides the gnomon in the ratio 

 of one to n ; this is evident by elementary geometry when Q is 

 brought close up to P, and so still holds as Q, separates again to 

 any distance along the curve. Here is the most evident repre- 

 sentation of the integration, with no apparent flaw in the 

 argument for the young Berkleian to attack, who is repelled by 

 the gritty approximation he meets in the course of a treatment 

 by summation, necessitating the lengthy abstract apologies of" 



