150 B. V. KOTLIAREVSKIT 



Here also, the result of integrating for ^3 was not dependent on x-^^; there- 

 fore it is the required value 61^. 



The position of the points for the second case, which is not given in 

 Fig. 4, presupposes that a < c^ < 2a. For this case the following expression 

 can be obtained for df^: 



dfm = jgoHa-d)^{2a-dr^ (17) 



On the right-hand sides of formulae (11), (12), (16) and (17) there is the 

 value d. It can easily be seen that 



where A^q is the increase in gravity between two neighbouring points of 

 observations. We determine Zl^o ~ ^10 (^' ^)- 



As above, we will seek this expression for the gravity curve in an appro- 

 xiinate parabola passing through three points. Since the positive branch 

 of the curve g{x), shown in Fig. 4, is identical in form to its negative branch, 

 to find the mean integral value of A^q = (g^—gi)^ it is sufficient to take 

 the range of integration from to 2 (remembering that Iq is taken equal to 

 unity). 



Here there can be two cases: (a) both points of observation (% and x^ 

 lie Avithin the Umits of the positive branch of the curve g{x) ; (b) the point 

 a;^ is within the limits of the positive branch while the point x^ is in the H- 

 mits of the negative branch (this case is not shown on Fig. 4). The values 

 of ZljQ will be different for these cases. We will designate these values by 

 (z1iq)i and {Aj^q)^, respectively. Dispensing Vith the intermediate calcula- 

 tions, we obtain: 



for 0<A;i<2-a 



iAi,),=glan{2-a)-2x,Y, 



for 2-a<:^i<2 



{Al,),=gl[{8-6a+a^)+2{a-^)x,+2xl]^ 



The required average integral value of the form A^q in the range 

 ^ ,Tj ^ 2 will be equal to half the sum of the following integrals : 



2-a 



2-a 



2-a 2 



j[f{^lo)idx,+J{Alo),dx,'^ 



