12S 



E. L. De Forest — Un symmetrical Laxo of Error 



dxdy 



fj ^'^zdxdy^ B 



dxdy 



ffy^zdxdy. 





(1') 



The diiferential equations (14) or (16) cannot, I believe, be inte- 

 grated in their complete form. But if we neglect the inequalities 

 ?/j and 7/2,. (14) reduces to 



d^z-\{d^^fi^)d^^_ -X ^ 



z nfi^dx^ 'y 



d.-h{d^^l3^)d-'^_ -y \ 



(18) 



z rili.^dy j 



These equations are of the same form as the one near top of p. 138 



in my article on the Ij nsymmetrical Probability Curve. That equa. 



tion was 



dy—{{b^-h-b^)d'^y —x 



y 



If we write 



it becomes 



kh^dx 



•lb 

 kb.^ (dx) '^ = b, kb^ (dx) ^ = — , 



S^KI)-(l)-=o, 



(19) 

 (20) 



(21) 



a linear differential equation whose exact integral is of a highly 

 transcendental form. (See Price's Calcidics, vol. ii, p. 652). But as 

 shown in my article, an appi'oximate integration can be effected, with 

 a comparatively simple result. In applying the method to (18), we 

 transfer the origin to another point by putting 



X f^ and y-—^-— (22) 



in place of it; and y respectively, and write new constants 



«.='w' "-=W$' ^=.</^,«'-)n h=«fiM,)^- (-) 



Thus (18) will rcdiice approximately to 



d^ dx, .,, ^. , <i^ dy . , . , , 



-^= — {a.-b^—\)—a.dx^ ■^=.^{n^ib^ — \) — a.,dy, (24) 



z X z y 



and integration gives 



a{^hi — \ <i.,-b., — \ -UiX—(hy . 



z=Cx y e . (25) 



To restore the two equations (24), we have only to differentiate 



(26) with lespect to ./• and y separately, and divide tlic results by <■• 



