1886.] Family Likeness in Stature. f>5 



Fig. 8. 



^ + ( *- ;/ w)* =C) 



a* b 2 ' 



equation (3) becomes 



(8) 



V'r > a 2 tan 

 or * = tan 0= 



« ^ fc 2 + a 2 tan 2 



and (5) becomes - = L _ ^ an ^ (9) 



e 2 a 2 6 2 



whence tan0_e 



tan0 6 2 v y 



If c be the probable error of the projection of p's whole motion on 

 the plane of xz, then 



c 2 =a 2 tan 2 0+6 2 , 



which is independent of the distance of p's line of motion from the 

 axis of x. Hence also 



tan0 a 2 (11) 



tan 6 c 2 



Problem 2. 



An index q moves under some restraint up and down a bar AQB, 

 its mean position for any given position of the bar being Q ; the bar, 

 always carrying the index with it, moves under some restraint up 

 and down a fixed frame YMY', the mean position of Q being M : the 

 movements of the index relatively to the bar and of the bar relatively 

 to the frame being quite independent. For any given observed posi- 

 tion of q, required the most probable position of Q (which cannot be 

 observed) ; it being known that the probable error of q relatively to 



VOL. XL. F 



