MULTIPLE ALGEBKA. 117 



already considered. It is to be observed that these multiple differ- 

 ential coefficients are subject to algebraic laws very similar to those 

 which relate to ordinary differential coefficients when there is a 

 single independent variable, e.g., 



da- dr _d<r 

 dr dp dp' 



dp dr _.. 

 dr dp 



In the integral calculus, the transformation of multiple integrals 

 by change of variables is made very simple and clear by the methods 

 of multiple algebra. 



In the geometrical applications of the calculus, there is a certain 

 class of theorems, of which Green's and Poisson's are the most 

 notable examples, which seem to have been first noticed in connection 

 with certain physical theories, especially those of electricity and 

 magnetism, and which have only recently begun to find their way 

 into treatises on the calculus. These not only find simplicity of 

 expression and demonstration in the infinitesimal calculus of multiple 

 quantities, but also their natural position, which they hardly seem 

 to find in the ordinary treatises. 



But I do not so much desire to call your attention to the diversity 

 of the applications of multiple algebra, as to the simplicity and 

 unity of its principles. The student of multiple algebra suddenly 

 finds himself freed from various restrictions to which he has been 

 accustomed. To many, doubtless, this liberty seems like an invi- 

 tation to license. Here is a boundless field in which caprice may 

 riot. It is not strange if some look with distrust for the result 

 of such an experiment. But the farther we advance, the more 

 evident it becomes that this too is a realm subject to law. The 

 more we study the subject, the more we find all that is most useful 

 and beautiful attaching itself to a few central principles. We 

 begin by studying multiple algebras; we end, I think, by studying 



MULTIPLE ALGEBRA. 



