from Equations of Coexistence. 429 



The above being " equations of coexistence," x is called " the 

 repeating term." 



If we suppose the equation 



CqX"- + Cj .r'"-l + Cr^X""-^ + + Cr = 



to be capable of being deduced from the two above, and, there- 

 fore, necessarily implied by them, this will be called " a Par- 

 ticular Derivative " from the equations of coexistence of the 

 ^th degree, (r being supposed less than m and 7i *, and the co- 

 efficients being rational functions of the coefficients of the 

 equations of coexistence.) 



There will be an indefinite number in general of such de- 

 rivatives, and the form involving arbitrary quantities which 

 includes them all is called " the general derivative of the r*^ 

 degree." 



Any " Particular Derivative," in which the terms are all 

 integral, numerically as well as literally speaking, is called an 

 "Integral derivative." 



That " Integral Derivative" of any given degree in which 

 the literal parts of the coefficients are of the lowest possible 

 ditnensions-f, and the numerical parts as low as they can be 

 made, is called the "Prime Derivative" of that degree. So 

 that there is nothing left ambiguous in the prime derivative 

 save the sign. 



The " Derivative by succession " is that particular deriva- 

 tive which is obtained by performing upon the equations of 

 coexistence, the process commonly employed for the discovery 

 of the greatest common measure and equating the successive 

 7'emainders to zero. 



To express the product of the sums formed by adding each 

 of one row of quantities to each of another row, we simply 

 write the one row above the other; a notation clearly capable 

 of extension to any number of rows, which would not be the 

 case if we spoke of differences instead o( sumsX. 



Theorem (1.) 

 Let hii h^, ... km) be the roots of one equation of coexistence, 



* This restriction upon the value of r is not essentially requisite, and 

 is only introduced to keep the attention fixed upon the particular objects 

 of this 1st Part. 



t Of course the dimensions of the coefficients in the equations of co- 

 existence are to be understood as denoted by the indices subscribed. 



X The wider views which I have attained since writing the above, and 

 which will be developed in a future paper, lead me to request that this no- 

 tation may be considered only as temporarj'. It would have been more 

 in accordance with these views to have used the two rows to denote pro- 

 ducts of differences than of sums. But a change now in the text would be 

 very apt to cause errors in printing. 



