VECTOR DIFFERENTIATION. 
75 
Assyrian harp. The operation so denoted was discovered 
by Hamilton, and he was led to the discovery by asking the 
question, What is the square root of Laplace’s operator, 
A + A + A? 
<W \ Stf ^ <Sz 2 ■ 
He reasoned that, since ix + jy •+* kz is the square root of 
— {x? + y* + z 2 ), the square root of the negative of Laplace’s 
operator is 
. 8 .. 8 
He perceived that the discovery would be of great impor¬ 
tance in the development of mathematical physics, and he 
indicates some of the first results; but the actual develop¬ 
ment has been carried out principally by Professor Tait, who 
on that account was happily designated by Clerk-Maxwell 
as “ the chief player on nabla.” 
The original definition of nabla, given by Hamilton, is 
purely symbolic and is as follows (. Lectures on Quaternions , 
page 610): “ Introducing for abridgment as a new character¬ 
istic of operation a symbol defined by the formula 
• d , • d , 7 d 
dx ^ dy dz y 
which is to be conceived to operate on any scalar or vector 
or quaternion regarded as a function of the three independ¬ 
ent variables x, y, z } we shall have generally the formula 
,.. , . , r n f dt | du , dv \ , • f dv du\ 
■ • / dt dv \ , 7 ( du dt \ 
^ \ dz dx) \dx dy J’ 
where t, u, v may denote any three functions of those vari¬ 
ables, x, y , 2 .” 
The definition given by Professor Tait is as follows: “ We 
commence with a form of definition of the operator y some- 
