562 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Notation. — The notation used in this paper will be that of Gibbs, 

 in which all vectors may be distinguished by being printed in Claren- 

 don type, while scalars will be in italic type. Thus, a would be a 

 vector, and a a scalar. The magnitude of any vector a will be denoted 

 by|a|. 



The scalar product, 



• (a^b^ + a^b^ + a^bj), 



of two vectors a and b will be denoted by a-b; and their vector 

 product, 



i (a^b, - a,b,) + j (a,b^ - a^b,) + k (a,b^ - a,b^), 



by axb, where i, j, and k are the unit vectors in the directions of a?, y, 

 and z, respectively. 



The symbol V will be used for the operator. 



( 



.3 .a , a 

 1 V- +Jh- + k — 



ox dy oz 



so that va is the gradient of scalar a, a vector; V -a is the diverg- 

 ence of vector a, a scalar ; and v xa is the curl of vector a, another 

 vector. 



The "mass" of a body must be understood to mean its mass as 

 measured on a system of axes on which it is at rest; and similarly the 

 density, which will be denoted by the letter p, will mean the limit of 

 the ratio of the mass in a volume element to the volume of the element, 

 with the above interpretation of the word mass. The mass of a body 

 will, therefore, be a measure of the amount of matter, or " gravita- 

 tional charge" in the body, which will be different for moving bodies 

 from the "inertia" usually understood by the word mass. 



Other Theories of Gravitation. — With this notation we may now 

 examine the results of various methods by which we might imagine the 

 changes of gravitational force to be propagated. 



First, let us imagine, according to what I shall call " Theory I," that 

 the changes of gravitational force due to any changes in the position 

 of matter take place at the same instant at every point in space. In 

 this case, if g is the gravitational force per unit mass at any point. 



g 



^■v///;* = -^///r''^'* 



where r is the distance from the volume element dr to the point at 

 which the integral is to be evaluated, x^ the unit vector in the direc- 



