468 Prof. Sylvester on the Properties of the Test Operators 



operators which are made to act on the functions themselves,, 



will give rise to r . first derivatives, which, energized in 



their turn, will be commutable inter se and with the original 



operators : the derivatives of the next order enjoying the same 



r -4- 1 r I 2 

 properties will be r . — -= — — ^— , and so on. Thus, as before, 

 & o 



we obtain the prime pertractive derivants of various orders, with 

 the difference that there are now several of such prime derivants 

 belonging to each order. Any function of these gives rise to a 

 compound-perfcractive derivant, the number of which is of course 

 unlimited ; these may be energized into operators, subject inter se 

 to all the laws of algebraical operation, and any function of one or 

 more of such compound-pertractive derivators will be equivalent 

 to some single derivator belonging to the same family. In a 

 word, the theory may be extended from the case of Monocephalous 

 to that of Polycephalous pertractive functions and operators and 

 their derivatives. 



I will conclude for the second time with the statement of an 

 expansion in a series which, as far as I have been able to ascer- 

 tain, is new to writers on the differential calculus, to which I 

 was led by applying the symbolical equation previously given in 

 a footnote to the operand a x . The equation in question may be 

 written as follows : 



from this I have been able to deduce by a mental calculation, 

 the steps of which I am unable to recall, a development which 

 would be exceedingly difficult to obtain from the method of 

 Maclaurin's theorem. I find 



+ W"(n + l)(n+2)(« + 8) + "•' 



where in general S ij7 - signifies the sum of the — ' — 4 ' 



products of the combinations of the numbers 1, 2, 3, . . . i, 

 taken j and^ together. This development may be easily verified 



in winch it will be noticed that the three last terms may be obtained (to a 

 constant factor pres) by operating with the sum of the three first upon the 

 sum of the three middle terms, or vice versa. 



