291 



terly Memoir on Surfaces of the Second Order, which has 

 been published by Professor Mac CuUagh in the Proceedings 

 of this Academy. (See Part VIII., page 484.) 



The length to which the present abstract has already ex- 

 tended, prevents Sir William Hamilton from oflfering on the 

 present occasion any details respecting the processes (analo- 

 gous in some respects to the calculi of variations and partial 

 differentials) by which he applies the principles of his own 

 method to investigations respecting surfaces and curves in 

 space, or to physical problems connected therewith ; he de- 

 sires, however, to mention here that, in investigations respect- 

 ing normals to surfaces, he finds it convenient to employ a 

 new characteristic of operation of the form 



(s.dp)-'.d = a, (48) 



in order to obtain from a scalar function of a variable vector 

 p, a new variable vector v which shall be normal to the locus 

 for which that scalar function is constant ; and that the fol- 

 lowing more general characteristic of operation, 



in which x, y, z are ordinary rectangular coordinates, while 

 ?,y, k are his own coordinate imaginary units, appears to him 

 to be one of great importance in many researches. This will 

 be felt (he thinks) as soon as it is perceived that with this 

 meaning of <1 the equation 



is satisfied in virtue of the fundamental relations between his 

 symbols i, j, k ; which relations give also, as another result of 

 operating with the same characteristic, this other important 

 symbolic expression, which presents itself under the form of 

 a quaternion : 



