1 58 MA THEM A TICAL NOTES. 



for by the ordinary equation of the tangent plane we have 



I u I m 1 mm dF m dF m dF 



x o ' #0 ' ^o " dx ' fy ' dz ' 

 F=Q being the equation to the surface. 



Again, if we seek to determine the surface, so that $ shall be 

 constant, i. e. to find the envelope of all the planes represented 

 by (1), we have (1), (2), (3), (4), as before, and in addition 



Thus, as before, ?/ y = / * =/ 3 , and the equation of 

 the surface may be got by integrating (5), and determining 

 the constant so that the result may coincide with (6). And the 

 identity of the equations connecting # , # , Z Q , and x, y, z, in the 

 two cases proves our proposition and its converse. Take as an 

 example the ellipsoid 



a* b* c 2 



=-, 



Therefore the tangent plane to any point of the ellipsoid 



2 T2 2 



makes 5 + 2 + 2 , a minimum with reference to any plane 



^o y" *! 

 passing through that point. 



2. To find the value of 



A A 



+&c ,. A 



when a t = a 2 = &c. = a. 



Let a = a + z y a = a + z &c. 



and so of the rest ; 



' A = fa k-,,).. 1 ...^-^ + K-^).. 1 ...^-^ + &c j 



+ &C. 



