SURFACES OF THE SECOND DEGREE. 93 



of the criteria. If we look at F'l, V^, Fs, and F^, we find that those 

 terms, and those terms only, which are multiplied by cosines of f, &c., 

 are of the first or third degree, with respect to any of the three sets 

 just mentioned. 



The case is very much altered when we consider any numerical 

 relation, however simple. For example, I give the condition which 

 expresses a surface of revolution, or a surface two of whose axes are 

 equal. If A and A' belong to the equal axes, a, a, &c. become in- 

 determinate; hence the numerators of the six equations (38), will, when 

 equated to zero, have a common root. Eliminate F - FA + FoA" from 

 the values of a* and (3y, &c. in (38), which gives 



.-_ XaCOsf-Z^ MiCO^ri-D iVgCOS^-iV^ ,^ 



AFa = r"=- = -7 T~ - y - V4b), 



a cos ^— a cos t] — o ccos^— c 



which does not admit of any material simplification. There are evidently 

 other ways of obtaining corresponding conditions from (38). I have 

 chosen this because the corresponding formulae have been given in the 

 case of rectangular co-ordinates. In this case, 



cos ^ = cos >; = cos ^ = 0, and Li = - l„, &c. 



whence, 



^ _ ^/ _ ^/ 

 a b c 



(See Mr Hamilton's Analytical Geometry, p. 323.) 



To apply the formulae (39) and (41), let there be two planes whose 

 equations, separately considered, are 



\'x+ fi'y+ i/'a + l =0 J 



but which together must be one of the varieties of equation (3). Let 

 new and rectangular axes be taken, the intersection of the planes 

 being that of x. Their equation will then be 



[c] z" + 2[a]yss = 0, 



