Notices respecting New Books. 225 



of given modulus for an assigned amplitude by deductions from this 

 equation. When F(^») or u is regarded as a function of its ampli- 

 tude, it is found very convenient to call sin <p the sine of the ampli- 

 tude of u, and to contract these words into the notation sn u ; and 

 similarly cos <p and \/(l — & 2 sm 2 <^>) are denoted by cnti and dau: 

 e. g. the addition equation given above would be written 



en (w + v) = en u en v — sn u sn v dn (u-\-v ). 

 There is no difficulty in deducing an enormous number of relations 

 between these functions, such as 



sn (u -f v)(l — k 2 sn 2 u en 2 v) = snwcnydnv + snvciiwtln u. 

 It will be seen that this formula closely resembles the formula for 

 the sine of the sum of two angles, which indeed it becomes if k=0. 

 The fourth chapter contains the development of the relations be- 

 tween the functions of the amplitudes and those of the sums, differ- 

 ences, and multiples of the functions. 



The third chapter contains : — the solution of a number of elemen- 

 tary questions, particularly those relating to the curves whose arcs 

 represent the integrals; the discussion of " the march " of the func- 

 tions, or their graphic representation ; some of the properties of 

 the complete functions (viz. the integrals when ^ = |7r), such as the 

 theorem enunciated by the equation 



EF+ET-FF=|7r; 

 methods of obtaining series for the complete functions when k is 

 either small or nearly equal to unity ; and the properties of the 

 " Grudermannian," i. e. the amplitude of the function F(0) when k 

 is unity. 



It has been already mentioned that a method of calculating ~F(<j>) 

 numerically might be based on the addition equation. Practically 

 speaking, however, the calculation is more conveniently effected by 

 transforming the integral with a given amplitude and modulus into 

 another with a different amplitude and modulus, and repeating the 

 process of transformation until an integral is found of assigned am- 

 plitude with a modulus not differing sensibly from zero. Chapter 

 13 is devoted to showing how this can be done, by means of 

 Landen's theorem, for functions of the first and second kind. The 

 chapter contains the original proof of Landen's theorem in an 

 altered notation. A proof of the same theorem is also given in 

 chapter 2, as a deduction from one of Jacobi's proofs of the addition 

 equation, viz. that depending on two circles. It may just be no- 

 ticed that this first course gives a view of the subject about equiva- 

 lent to that attained by Legendre shortly after he had begun his 

 labours, though many points are here worked out systematically 

 and in a new notation (particularly all that relates to the functions 

 sn u &c.) ; and some points are new, such as the proof of the double 

 periodicity of these functions, the properties of the Gruderman- 

 nian, &c. 



These chapters occupy about a third part of the volume, and mil 

 not offer any serious difficulty to the student, though he will have 

 to dwell upon them sufficiently to become accustomed to the nota- 

 tion, much of which will be new to him. The difficulty increases 



PMl. Mag. S. 5. Vol. 3. No. 17. March 1877. Q 



