TO THE COMPARISON OF SEVERAL MAGNITUDES. 63 



By successive substitutions we arrive at a value of A, in terms of R, S, T, 

 which may be conveniently represented by the equation, 



A = yn . R + On . S + ^n . T , 



and our business becomes to discover the law of formation of the functions fn, 

 On, \fm. 



Continuing the operation one step further, we have 



B, = P„ + 1 .S + g B + 1 .T + r n + , . U , 



and substituting this value for E in the preceding equation, 



A = {p„ + i- <pn + On] S + {q n + x . <pn + xjjn} T + r n + l . <pnU , 



wherefore the law of successive formation is contained in the three equations — 



<p(n + 1) = p n + 1 . <pn + On , . . . (1), 

 0(n + 1) = q n + 1 .Qn + xjtn, . . . (2), 

 xjj(n + 1) = r n + 1 . <pn, .... (3). 



Now these forms hold good for every value of n, wherefore 



xjjn = r n . <p(n - 1) , 

 and consequently 



0(n + 1) = <p n+1 . <pn + r n . <?(n - 1) , 

 whence 



On = qn . q(n — 1) + ;•„ _ x . <p(n — 2) , 



so that the equations (1), (2), (3) become 



<p(n + 1) = p n+1 . pi + q n . <?{n - 1) + r n _ x . <p(n - 2) , (1), 



0{n + 1) = q n + 1 . <pn + r n . <p(n - 1) , . . . (2), 

 xjj(n+ 1) =r n + 1 .pi, (3). 



By means of these formulae we can construct the series of functions tpn 

 independently of the others, and thence we can readily deduce the progression 

 On ; as for the third progression \pn it is, in the usual case of r = 1, a mere 

 transcript of the progression <pn moved one step back. The arrangement of the 

 work is very simple, and may be best studied from a numerical example. 



The arrangement of the intervals in music has to follow the natural sub- 

 division of a vibrating column, and so must be made, primarily, to suit the 

 ratios 1 : 2, 1 : 3, 1 : 5, and their compounds. If, then, it be proposed to tune a 

 musical instrument so as to permit of transposition from one key to another, the 

 ratio represented by the smallest interval on it must be contained exactly, or 

 very nearly, in each of these three ratios. Therefore the arrangement of the 

 gamut on an instrument of equable temperament must be obtained by a com- 



VOL. XXVI. PART I. K 



