J oLY— Integrals depending on a Single Quaternion Variable. 15 



epoch and p as the vector to a variable representative point at the 

 time t. For a single integral taken between fixed limits, the repre- 

 sentative point is obliged to leave a fixed position at a given time and 

 to reach another fijxed position at another given time. The path it 

 describes and the rate at which it traverses that path are fully 

 specified by the mode of passage. 



Ai't. 11. — To give an illustration, take the case of the scalar 

 integral 



Q = /S(^4o-)(di + dp) = jUdt + J So- dp. (49) 



"We have seen (18) that the conditions that this integral should be 

 independent of the mode of passage are 



^+V^=0, VVo- = 0. (50) 



ot 



The form of these equations suggests an example from fluid motion, 

 so we suppose o- to be the velocity of the fluid which the second 

 condition requires to be irrotational. To see what interpretation 

 we may assign to the scalar £, we write down the equation of 

 motion, the suffix denoting that o-q is not operated on by V, (compare 

 the Appendix to vol. n. of the ''Elements of Quaternions," p. 547), 



6- = ^ - So-oV-o- = - VP V« (51) 



at c 



in which P is the potential of the impressed force, c the density 

 and p the pressui'e. But we have identically 



Yo-oYVo- = So-oV-o-- VSo-oO- = So-oV.o- + Vi-To-- = 0, (52) 



and therefore 



^= _vP-ivi>-iVTo-^ (53) 



at c ~ 



Thus we may take 



U = F+{-dp + iTa^ (54) 



c 

 so that £ is the energy of the fluid per unit mass. 



In general in the case of fluid motion when £ is the energy per 

 unit mass and o- the velocity of the fluid at the representative 



