34 THE MECHANICS OF THE EARTH's ATMOSPHERE. 



Hitherto, almost exclusively, only those cases have been treated 

 where not ouly the forces X, ¥, Z, have a potential Fso that they can 

 be expressed in the form, 



X=^r=^,Z=^ (la) 



but also where a velocity potential cp can be found so that 



u = ^, V = ^^, ,r = ^, (16) 



The problem is thereby greatly sim|)lified since the first three of 

 equations (1) give a common integral equation from which to find p 

 after we have determined cp in accordance with the fourth equation 

 which ill this case takes the form 



and which therefore agrees with the established differential equation 

 for the potential of magnetic masses that lie outside the space within 

 which this equation holds good. Moreover, it is known that every 

 function q) that satisfies this last differential equation within a simply 

 connected s^jace,* can be expressed as the potential of a definite dis- 

 tribution of magnetic masses on the boundary surface of the space as 

 I have stated already in the introduction. 



In order that we may be able to make the substitution required in 

 the equation (1&) we must have 



jj-^=^^ fz-^j-^^ yx-yz=^^ (1^-^ 



In order to understand the mechanical significance of these last 

 three conditions, we may imagine the change that any infinitely small 

 volume of water experiences in the elementary time dt to be com- 

 pounded of three different motions : (1) a motion of transference of the 

 whole through space: (2) an expansion or contraction of the particle 

 along the axis of dilatation, whereby every rectangular parallelopipe- 

 dou of water whose sides are parallel to the principal axis of dilata- 

 tion remains rectangular while its sides change their lengths but re- 

 main parallel to their original directions : (3) a rotation about some 

 temporary axis of rotation having any given direction, which rotation 

 can by a well-known proposition be always considered as the resultant 

 of three rotations about the three coordinate axes. 



*In manifold-connected-sijaces 9 cau have several A'alues, but for such many val- 

 ued funotioDs as satisfy the above ditferential equations the fundamental proposition 

 of Green's theory of electricity no longer holds good (see Crelle Journal, Xhiv, p. 360, or 

 "The Mathematical Papers of the late George Green"), and therefore fail also a 

 greater part of the propositions resulting from this which Gauss and Greeu have 

 demonstrated for the magnetic potential functions, which functions are in their very 

 nature always uui-valued. 



