1052 HON. LORD M'LAREN ON SYSTEMS OF 



whence 1/r" is found by summing the terms. But for purposes of computation the 

 formulae of the preceding paragraph are preferable, because they contain only half the 

 number of trigonometrical quantities that are contained in the polar expression. 



Where there are three variables, and it is desired to obtain values of a radius vector in 

 terms of 6 and the spherical angle <f>, the computation may also be simplified by making- 

 use of cylindro-polar coordinates. In this system r is the radius vector in the plane of 

 XY ; thence x = r cos 6 ; y = r sin 9 ; z = r tan <f>. Each term of x, y, z then contains at the 

 most only three trigonometrical quantities to be computed, instead of Jive, as in the 

 ordinary spherical system, and the angles 6 and <j> are the same. 



The spherical radius vector, if required, can be afterwards found by the relation, 

 spherical radius-vector = rsec</>. The equation of three variables transformed to 

 cylindro-polar coordinates is of the form 



cos"0+ JA cos" " 2 0.sin 6 + A x cos" - 3 0.sin 2 + . . . }tan 0+ {cos"- 3 # . sin0+ . . .[ tan 2 ^ 

 + . ■ . + sin"0+ tan"</> = l/r n . 



Examples of solutions effected by transformation to polar and cylindro-polar coordin- 

 ates will be given in the sequel. 



5. Solution of certain Homogeneous Equations by the introduction of a New 



Variable. (Case III.) 



It is only in the case of homogeneous equations that the n th root of the arbitrary 

 term is a parameter or value of x when y = 0. In all other cases the parameter is 

 determined by an equation in x or y (as the case may be), which in the case of the 

 higher degrees can only be solved by approximation. Hence the method of homogeneous 

 variation is not directly applicable. In applying the principle of homogeneous variation 

 to functions which are not homogeneous, we must consider the function of two variables, 

 as a particular value of a function of three variables in which z has become unity. Thus, 

 if we suppose a surface to be represented by an implicit homogeneous function f(x, y, z), 

 a plane section, parallel to the plane xy and at a distance from the origin z = 1, will be 

 represented by the heterogeneous equation formed by the disappearance of the quantit}^ z. 



In order to solve an equation of the form u n + u n _ i &c. = 0, we must first restore it to 

 the homogeneous form by introducing such powers of z as will make the equation homo- 

 geneous, and then endeavour to reduce z to unity by homogeneous variation. 



Consider the two following equations, in which the brackets include terms of the 

 same degree in x and y : — 



[x +A l x n -hj±...±A„y"\ +{.x"- 1 + B i a;"- 2 ?/ 2 ±. . . ±B >l . 1 y n - 1 } + -jx" - 2 + &c. } = 0. . . (1) 



j^ + A^-^i. . . ± Am") + l^^+B^-^db. . . ± B„_!,r !}£+ {£"~ 2 + &c.}f + &c. = (2) 



The first form is a thoroughly heterogeneous equation, containing terms of degrees 



