392 Mr. H. T. Flint on Integration Theorems 



These five equations determine the five quantities necessary 

 to fix the plane, which is therefore definite if Y 2 s:/3 is known. 

 It is easy to show that Y 2 yS written in full is : 



where the A's denote the components of the original plane. 

 It is to be noted that the coefficients Ay Z and A XM , and simi- 

 larly for the other pairs, change their places in passing from 

 a plane to its normal. We have tacitly assumed in this 

 expansion of Y 2 yS that Y 2 a/3 is also a unit plane. If it is 

 not, we have only to multiply throughout by a constant — the 

 reciprocal o£ the tensor. 



Thus if a plane is represented by kY 2 , where k is a scalar 

 constant, the tensor of the plane and V 2 is the unit vector 

 defining it vectorially, any plane normal to it is //Y 2 ', where 

 k' is the tensor of the latter and Y 2 ' is obtained from Y 2 by 

 interchanging the coefficients in the above manner. 



Functions enter into analysis which have the structure of a 

 vector area but do not satisfy the condition (2*5). They are 

 defined by 6 quantities, and we may write such a function as 



P 2 = i 2 U ? yz + hh^ zx + i^xy + iiiiPxu + hk^yu + hhYzu, (2'7) 



and there corresponds to this the reciprocal function : 



T 2 '=.uj B V xu + + +hi 4 P**+ + (2-8) 



But in this case the relation 



V 2 P 2 P 2 '#0, 



as in the case of the product of vector areas when these are 

 normal. 



We shall return to this point later. 



It is to be noted that we can always express P 2 in the 

 form : 



P 2 = / L Y 2 «/9 + //Y 2 'a/5. 



k ! gives the additional arbitrary constant. 



This should be compared with Sommerfeld's discussion, 

 § 1 of vol xxxii., already referred to. 



If a vector area be split up into elements so that each 

 element projects into rectangles on the coordinate; planes, we 

 write for it : 



dv — i&zdy dz + i 2 i\dz dx + i x i 2 dx dy 



+ itfidx du + ifadydu + i^dzdu. . (2*9) 



This element is a vector like P 2 . 



