Researches in the Tlieory and Calculus of Operations. 235 



is reduced from R. to R', and the lunar emanation from S to 

 S'. The Moon arriving at the zenith, the meridianal belt of 

 the Earth's surface in view receives an additional pressure 

 from the lunar emanation, proportionate to the difference 

 between R,R' and S,S', or to their sum when the Sun is on 

 the same line and side of the Earth. This added pressure 

 depresses the fluid portions of the Earth at A, and liberates 

 sufficient force at the opposite point A' to produce a corre- 

 sponding elevation of the waters at that point. As the Moon 

 passes off the meridian, the pressure departs westward, re- 

 action of the lately depressed portion of water supervenes 

 and is reinforced by the "subsidence of the waters of the 

 opposite side of the globe, precisely in the manner of other 

 forms of oscillation. The centrifugal force of the globe con- 

 tributes of itself to increase the elevation, and of course to 

 decrease the depression of the waters ; and thus the daily 

 retardation of the moon's orbital motion behind the earth's 

 rotative motion keeps up a perpetually circulating wave 

 from east to west around our planet, the magnitude of the 

 effect being constantly modified by the action of the Sun. 



4. By the use of geometrical multipliers, the principle of 

 areas, and the so-called laws of Kepler, may be constructed. 



Mathematical symbols express relations of operations per- 

 formed by forces in space and time. Addition isan operation, 

 and multiplication is but an abbreviated method of perform- 

 ing a series of additions. 



Let 1^ denote the unit of mass, the multiplicand or body 

 operated upon, represented say by the sphere (fig. 4); and 

 let 1, 1,, 1 K respectively denote the units of time, of space 

 (rectilinear distance), and of velocity (rate of motion). As 

 the magnitudes of these units are arbitrary, they may be 

 chosen all equal to the line 1 (fig. 29). The unit of ve- 

 locity l x being the multiplier, the operation l„xl„ is ex- 

 plained by the application of a force (say an impulsion) to 

 the multiplicand 1„ placed at the point 0, which will carry 

 it from to the point 1 in the unit of time l, f giving the result 



