Compound Law of Error. 213 



From (7) and (8) we obtain 



„__J_ /_£_ V_ . JJ JL. JL JL\ (9) 



' sjkm X \sjV sjin k V km?' kW mV * W 



Put z 1 for the first approximation which has already ^been 



found, viz. : 



1 -(? + ^\ 



z-i = = e v- 2 * 2 »»/. 



2 sjkm 



Put z=Zi(l + <d). Then for @ we may substitute with 

 sufficient generality 



A; 5 km 2 k 2 m rri* 



where the 0's are functions of x and y ; since, the coefficients 

 in (9) being by parity of reasoning with that employed in the 

 case of a single variable* small, second powers do not 

 appear in the expression for z. Since z reduces to z x in the 

 case of symmetry, 6 must be null. For the other unknowns 

 we have from equations (3), (4), (5), and (6) , neglecting small 

 quantities, 



1 



1 d z z x 1 /} _ 1 & 



with corresponding equations for 4 and 3 . Performing the 

 work we have for the asymmetrical probability-surface, 



z — 



V2& 2m) 



2ir si km 

 \ 2# */k\ ok) 2k\/m\ / m\ h) 



2^/k *Jk\ m) 2 m i v^\ 3m;;- ^ ' 



By construction, 2 satisfies equations (3), (4), (5), and (6). 



From the form of the expression it is evident that it.satisfies 

 equation (9), and therefore equations (7) and (8). By actual 

 trial it is found to satisfy equations (1) and (2) ; as may be 

 seen by breaking up the expression into five terms, and ob- 

 serving that each term separately satisfies those equations. 



We might also have proceeded by obtaining general solu- 

 tions of equations (1) and (2) in the form of series, after the 



* See the preceding article, Phil. Mag. 1886, xli. p. 90. 



