OF ARTS AND SCIENCES. 



399 



function, and {,j, I; &,c., are connected by some simple equation. This 

 solution can be developed into the form 



F {xi -\-ui-\- zh -f- &c.) = Mi -\- Nj 4- Pk + &c. 



in which M, N, P, &c., will be functions of ar, y, c, &c., and each of 

 them is a solution of the given equation. Thus in the case of Laplace's 

 equation for the potential of attracting masses, the vids must satisfy 

 the equation 



The algebra (a') of which the multiplication table is 



may be used for this case. Combinations t\,j\, k^ of these vids can be 

 found which satisfy the equation 



and if the functional solution 



-^(^^*i+.vyi+~^-i) 



is developed into the form of the original vids 



3n-\-Nj-\-Ph. 



M, N, and P will be independent solutions, of such a kind that the sur- 

 faces for which N and P are constant will be perpendicular to that for 

 which Jf is constant, which is of great importance in the problems of 

 electricity. 



THE USE OF MIXED ALGEBRAS. 



It is quite important to know the various kinds of pure algebra in 

 making a selection for special use, but mixed algebras can also be used 

 with advantage in certain cases. Thus in Professor Clifford's biqua- 

 temions of which he has demonstrated the great value, other vids can 

 be substituted for unity, and his new vid, namely their half sum and 

 half difference, and each of the original vids of the quaternions, can be 

 multiplied by these, giving us two sets of vids, each of which will con- 



