5S6 Messrs. Cowley and Levy : Method of Analysis suitable 



For a given form of R and definite end conditions the 

 nature of the solution of this equation depends only upon 

 the value adopted for C, and ail the members of the class 

 are included among the one-fold values of this class variable. 

 In fact, if the. differential equation representing any physical 

 problem is thrown into the non-dimensional form, one or 

 more class variables, themselves non-dimensional, will be 

 derived in the manner shown above. The differential 

 equation of the steady two-dimensional flow of a viscous 

 fluid, for example, when thrown into the non-dimensional 

 form 



where f represents the vorticity, V the speed of the moving 

 body, I its length, and v the coefficient of viscosity, indi- 

 cates that the properties of the motion will centre round 

 a consideration of the modifications in the solution as the 

 non-dimensional quantity Yl/v varies. 



The general solution of differential equations involving 

 such a class variable satisfying certain boundary conditions 

 which may themselves involve the class variable may be 

 written in the form 



/(.,:. y,C) = (1) 



The ordinary power series solution presents (1) in the form 



a + a x x -f b x y + a 2 x 2 + b 2 y 2 + c 2 xy +. etc. = 0, . (2) 



where of course the quantities a, b, etc. are functions of the 

 class variable C. In this form each of these coefficients 

 may be more or less complicated expressions of C, and the 

 whole expression is not in a form suitable for investigating 

 the relative properties of the members of the class, although 

 it may be convenient for other purposes. 



There is, however, another method of presentation of (1) 

 more suitable from the present point of view, viz. : 



X + Xic + X 2 c 2 +etc. = y, .... (3) 



where X , X l? etc. are now functions of x. The investi- 

 gation of the properties of the class and selection of a 

 particular member satisfying certain optimum conditions 

 can then be easily derived from this expansion. Whether 

 or not the solution can be thrown into this form in general 

 is a matter which cannot be discussed here, but w r ill clearly 

 depend upon the form of the differential equation and the 

 boundary conditions. Each case will require to be treated 

 on its merits. 



