Notices respecting New Books. 239 



machinery, will look at the question in a reverse order. This is 

 the outlook contemplated by Professor Perry. He takes a few 

 well chosen harmonic terms at the beginning of a Fourier series, 

 and examines the shape of the resulting graph, to see if it is a 

 good imitation of the mechanical result under examination ; in a 

 problem such as machinery balance or electrical surging. 



To raise the peak of a curve and to sharpen it if too blunt, 

 an odd harmonic would be introduced; the effect of the even 

 harmonic is felt in enlarging the haunches of a wave-arch. 



The figures in Chap. VII will give a lead in this treatment, 

 how to smooth out ripples if too prominent. 



Starting with an arbitrary Fourier series, such as on page 3, 

 and denoting it by 



i« + .... -\-a p cos j)irx /I -^-b p sin jj>7j\.i7/£-f-. . . . =f(.v)=y, 



then, in the interval l>.v> — Z, the coefficient a p is the average 

 y cospTrx/l, and b p is the average y sin f-KXJl ; and so Fourier's 

 theorem is stated without using the language of the Integral 

 Calculus, in a form that appeals to a practical mathematician ; 

 and the operation of averaging can be carried out mechanically 

 by wrapping the curve* of y round a cylinder, once, twice, 

 thrice, .... 



Simple statement of this kind is required in an appeal to t! e 

 interest of the beginner. 



In the applications, the a and b coefficients appear usually as 

 rational numbers, as for instance in the representation of 

 ?/ = ] — x. But the Jacobian zeta function, zn xK, is penultimate 

 to this curve, as >; — >1 ; the curve, a straight line, is imitated 

 without any ripples or discontinuities, and the Gribbs phenomenon 

 of p. 268 is absent. 



So too the curves of sn#:K or en xK may be investigated in the 

 penultimate form of tc — ->1, in their representation of broken 

 straight lines. Here the a and b coefficients are transcendental 

 numbers. 



This is the method of the average, described and resumed in 

 § 95, but a further discussion is inaugurated, as the statement 

 of the average is considered quite incomplete and inconclusive by 

 the author; and Dirichlet's conditions of §91 are to be satisfied 

 in the interval, leading to very delicate and abstruse consideration, 

 in which the author spreads himself to his own delight. 



He will not allow a proper comprehension of Fourier's Series 

 and Integral without a knowledge of what is involved in the 

 convergence of infinite series, and integrals; and the Definite 

 Integral is treated in Chapter IV from Eiemann's point of view. 



The volume is called an Introduction because the original work 

 is to be divided into two parts, the secoud part devoted to a 

 mathematical discussion of Heat Conduction, where the most 

 interesting applications are to be found, sequel to Fourier's 

 Theorie de la Chalear, of 1822. 



