124 Mr. F. L. Hitchcock on the 



single differentiation upon the given function, and trans- 

 forming to suit the purpose in view. The fundamental 

 formula for the work is (27), which contains the others. 



6. A full discussion of the general case, when /V is any 

 function of V, not necessarily linear, would exceed the limits 

 of this paper. A few illustrations of the meaning and use of 

 such operators must suffice. 



The simplest function of the second order in nabla is the 

 familiar V 2 - Another is SctVStV, which implies successive 

 differentiation of the operand along two different directions 

 a and t. The most general scalar function homogeneous and 

 of the second order in nabla is of the form SV$V, where </> 

 is any self-conjugate linear vector function. 



Vector functions of nabla depend on the properties of 

 vector functions of a vector. No general theory has been 

 worked out except for linear functions. In a future paper I 

 hope to show that a vector function homogeneous or the 

 second degree in a vector X may be put in the form 



V(f)\d\ + \$*\ 

 where <$> and 6 are linear vector functions and a is a vector. 

 Then the most general vector function homogeneous f the 

 second order in nabla would be of the same form, writing 

 V for \. 



We may, however, write down at once equations like (27) 

 for any differential operators whatever. Thus if d' and d" 

 are symbols of two independent differentiations, we shall have 



d'd , 'Fq=f 1 (d'q,d"g), .... (32) 

 in which f x will be a function linear in each of the two 

 differentials. Then if /(V, V v ) is any operator homogeneous 

 and of the second order in nabla, the differential nature of 

 nabla enables us to write 



/( V, V") . ¥q =/(V, V") .Mq<, q"), . . (33) 

 in which V' acts on g' and V /; on q" . Since d' and d" are 

 independent, f x is symmetrical in its two operands, a property 

 not in general possessed by /'in the two nablas. One or both 

 nablas will usually act on the constituents of/. 



7. The commutation of these differential operators is a 

 matter of importance both in Physics and in Mathematics. 

 Tait gives a proof of the celebrated theorem that vortex 

 motion cannot originate in a frictionless medium, by 



commuting kt with YV *• Commuting two ordinary 



differentiations along different directions gives rise to the 



so-called " Klammerausdruck," due to Jacobi, and important 



* 'Quaternions/ 3rd ed., Art. 513. 



