552 Mr. G. B. Jerrard's Reflections on the Resolution 



all the roots of the equation for y^ y^^ remaining fixed ; the 

 one will be said to be the complement of the other : and either 

 of these functions with the letter c prefixed to it as a charac- 

 teristic will express symbolically the complement of that func- 

 tion. 



12. It may now be seen, either from multiplying y^, y^, . . y, 

 successively by a^, or from the law of the indices in the equa- 

 tions (b.), that the ten equations, 



«/. + y^ - (' + •') y. = 0, 

 yu + yy-{^ + *') yfi = «» 

 2^/3 + 2^5 -(» + *')yy = o, 

 yy + y.-{^ + yi = o, 

 y^+y.-i^ + '^) ys=o, 



c{yy + y, -0' + »';ys) = o, 



where the indices in each vertical column follow the same 

 order of succession as in the cycle 



a 



(d.) 



(e.) 



cannot, whilst a, /3, . . e remain unaltered, conduct to more 

 than one set of values for p^ and pg. 



But before discussing the equations (d.) and (e.), I proceed 

 to consider properties of functions in which the elements are 

 supposed to change places among themselves. 



Section II. 



13. Let X represent a function of ti independent quanti- 



ties ^„ a:^, ^„ . . ; then X (^^ ^ ^^ .) or simply X (^ ^ ^ .) 



will express, according to a known notation, that in the func- 

 tion X the quantities x^., a^^j, w^j . . have been changed into 



