TRANSACTIONS OF SECTION A. 487 



It will also be noticed that, b being arbitrary, the integral (2) is equivalent to 

 two conjugate integrals, and may be more completely written thus, 



M = e ^(Acos«i; + Bsinrt7). 



The generality of this integral is due to the circumstance that the two lines OA, OB 

 (on which |, ij are the projections of »•) are perfectly arbitrary as to their direc- 

 tions in space, while ^, rj are entirely dependent, for their values in terms of .r, t/, 

 z, on the positions of OA, OB. 



Every different position of OA, OB, or of either of them, will give a special 

 integral, though every such special integral will be of the common type (2). Each 

 special integral will have its own arbitrary (or special) constants in the place of A, 

 a, b ; and any of such special integrals may be formed by addition into a group, 

 which group will be an integral of equation (1). 



There is no limit to the number of groups, but every gTOup will be composed of 

 integrals of the type (2) ; and it is in reference to this property that we designate 

 (2) the general integral. We may mention that^ = ?• cos AOP and ?; = ?• cos BOP, 

 and these cosines are easily expressed in terms of .r, y, z, and the arbitrary angles 

 which OA, OB make with the coordinate axes. 



Pnqy. B. — If now a third line OC be drawn at right angles to both the lines 

 OA, OB ; and if f be the length of the projection of r on OC, then will the 

 following differential equation hold good akvays, i.e. whatever be the positions of 

 OA, OB, OC!, 



dru (Pu dru _ ^ 

 d^^ dif ^ dC ~ ' 



from which it follows, that if we possess any integral F {x, y, z) of the equation 

 (1) we may write in this integral, instead of x, y, z, the values of |, r}, ( in terms 

 of X, y, z; and the integral F, though much changed thereby, wall still be an 

 integral of the equation (1). 



As a very simple example we may mention this, that if F (x, y, s) be an 

 integral, so likewise will F (.r cos a + y sin a, y cos a — x sin a, z) be an integral of 

 equation (1), though we have written .r cos a + y sin a and y cos a — x sm a instead 

 of X and y\ and a = is an arbitrary constant. And, thus, if the integral F 

 (,r, y, s) be only a farticvlar integral this introduction of an additional arbitrary 

 constant a, Avhicli it did not possess before, wiU advance it a step towards 

 generality. 



Pi-op. C. — The independent variables of equation (1) have usually been changed 

 by assiuning two angles, 6, (p, such that x = r sin 6 cos (f),y = ?• sin sin (f), and 

 z = r cos 6. It is somewhat more convenient in the work of integration to change 

 the angle 6 for its complement to a right angle. Thus we shall make the following 

 change of independent variables, 



X '= r cos 6 cos <p,y = r cos ^ sin <^, s = r sin 6. 



The transformed equation on these assumptions is 



2 d-u . n du , dru . /■ du .-.^ d-u ^ 



1 r^+ "r -7- + rrr:, - tau (9 — - + scc -6 —~ = 0, 



dr^ dr dff^ dO d(f)^ ' 



and of this the followmg is the general integral, 



M = Forces ^. e**^) + — / (r sec ^ . e"*^) (3) 



i is defined by the equation «'^ = — I, and F, / are arbitrary functions. 



The form of the transformed differential equation shows that -— - is also an in- 



dcf) 



tegral of it. Hence we may replace the second term of (3) by its differential 



coefficient with regard to (f) ; so that we may present (3) in the following form : — 



M = F (r cos e . e**^] + € ~*'^ sec 6f(r sec 5 . e~"^V , . . (4). 



