1050 HON. LORD M'LAREN ON SYSTEMS OF 



If be a homogeneous function of the n th degree in r variables, and ^ a homogeneous 

 function of the (n— p)' A degree in r variables, the equation 



(pyX-y, X^i • • .,X r ) = \jr\XyX^,. . -,X T ) 



has the solution 



1 V 0(« 1; ...,a r ) 



x - a „/ ^(ai, ••_•._%) 



r V^ a r ) 



x 3 — 



The cases here examined evidently include the following forms : — 



F n (x, y) = Ax n ~ p ; F n (x, y) = Ax r,+p 

 and 



F n {x,y) = z", 



where z p is a soluble function of other quantities, whose numerical value can be found 

 and stated as a power of z. 



4. Another mode of Solution; viz., by expressing each of the r Variables in terms 



of i—l New Variables. 



(1) Where there are only two variables X and Y, we have the relations Y = Xtan0 ; 

 X = Y cotan 6 . from which by substitution and division we may at once write the 

 transformed equations of the homogeneous function f(x, y) n = 1 , 



1 +a 1 tan0 + a 2 tan 2 -f . . . =v^ ■ ' ' ^ 



cotan"#+ +a 1 cotan n-1 # +a 2 cotan"" 2 + • • • = y^ • • • (^) 



Supposing a series of values of X to be formed from (1), and tabulated for the argument 

 6, then the column of values of Y is found by adding to each value of log X the cor- 

 responding log tan 6. 



(2) When there are three or more variables a, /3, y, &c, they may, in like manner, 

 be all expressed as functions of one of them, a, and new quantities. For this purpose, 

 assume /3 = atan<£ ; y = atani/;, &c, or more generally, (3 = la; y = ma ; 8 = na, &c. (3) 



Substituting these values, and dividing by a", the transformed equation will then 

 consist of a series of powers of I, m, n, &c, equated to l/a". Values of a may then be 

 directly computed for any arguments or assumed values of I, m, n; and the other 

 quantities, f3, y, S, &c, are formed from (3). 



The manner of doing this is shown by the following examples. (1 ) Let the equation be 



x*+xyz 2 +3xYz i +y* = w s = l ( 4 ) 



X is the quantity of which values are to be directly found ; 6 is the angular coordinate in 



