90 Letter from Mr. Ivory to Protessor Airy. 



make out any clear demonstration, to which it should be im- 

 possible lo oppose any objection, by a process of investigation 

 depending upon very delicate considerations, I chose to found 

 my reasoning, in the letter I addressed to you, on the nature 

 of the development. In that letter I have expressed myself 

 rather unguardedly with respect to M. Poisson's theorem : but 

 it would serve no purpose to enter upon any explanation of 

 these points ; because the demonstration I have now given is 

 equally clear and simple, and ascertains with precision the 

 nature and extent of this analytical theory. 



As it is now proved that the functions which come under the 

 method of Laplace are not entirely arbitrary, we have next to 

 inquire, how we are to distinguish them. And first we may 

 suppose that y is a function of three rectangular coordinates, 

 that is, of cos fl', sin fl' sin \|/', sin fl' cos vp'. On the surface of 

 the sphere conceive a spherical triangle of which 6' and fi are 

 two sides including the angle -^ — ■\i' ; then if y be the third 

 side, we shall have ^j = cos y. Further, let (^ be the angle of 

 the same triangle contained by the sides fl and y : and, by the 

 rules of spherical trigonometry, every one of the three (juan- 

 tities, cos 6', sin 9' sin ^', sin 8' cos vf/', may be expressed by a 

 linear function of cos y, sin y sin $, sin y cos ip ; and conse- 

 quently, y may be converted, by substitution, into a rational 

 and finite expression of the three latter quantities. But it 

 is easily proved that every such function is reducible to this 

 form, viz. 



y= ¥{p)+ v'T=]o^M, 



F being the mark of a rational function and M an algebraic 

 quantity having a finite value. Now y is what y becomes 

 when S' = (3 and 4>' = 4'? that is when ^j = 1 ; wherefore 

 7/ = F ( 1 ). Consequently, 



y-3/=F(i;)-F(i)+ ^/T^^.M. 



This expression is divisible by i/ I — p ; and therefore the 

 method of Laplace applies to rational and finite functions of 

 three rectangular coordinates. But, strictly speaking, this is its 

 whole extent : because we have no means of ascertaining that 



y — 3/ is divisible by \/ 1 — p, or of proving that the quotient 

 is a finite quantity, but the actual performing of the transfor- 

 mations I have shortly described. There is no doubt an ex- 

 tensive class of functions, not originally in the form of rational 

 expressions of three rectangular coordinates, but which may 

 be reduced to such expressions by converging series, as I 

 showed in my former letter to you ; and it will be admitted 



that 



