58 
Prof. OrNstei, who has pointed out to us this possibility was 
kind enough to give us some formulas, with the help of which we 
were able to calculate the angles required between the planes of | 
the semicircular canals and those of the otoliths. 
We once more want to render thanks to Prof. Ornstein, very 
much appreciating the trouble he took in introducing this matter 
to us. 
This method had the great advantage for us, that the results we 
first obtained by means of descriptive geometry, could be compared 
with the results got from the formulas. Through this we disposed 
of a welcome means for controlling the accuracy of our drawings. 
It lies beyond our reach to enter into details on the derivation of 
the formulas used here. Suffice it to represent the method in short: 
Formula for the calculation of the angle of two planes. 
Let the courdinates of the three points in the first plane be 
Bis Yi» 2,- ÌÎ point. 
EE a point. 
Bt Var 215 point. 
Secondly calculate 
Y, (2,—2;) it (¥;—4;) es (Wy. 232,93) = A, 
2, (z, —a,) a wv, (z,—2,) = (z, Ly, Z,) = As 
5, (Ya-Ya). + Yr (a) + (wa Ys Ya A0) = Ay. 
Let the coordinates of the three points in the second plane be 
Bia fis 188 point. 
Er Vin es AE paint. 
Ban Janes AT paint. 
Thirdly calculate 
ĳ (ei?) i eg Bie al he (Hs 202, Ya) =A,’ 
2, ii) + @,' (2; —2,) Li (ag8, > &425) = As 
te (ts HN dU hes Dieke (@.'Ys —Ya%s) = Ay. 
So, when Q is the angle of the planes 
AA! AAE AAS 
Cos. Q =—= wgn se 
VE = a aS =i A,’) (ay dr AS ds A 
The coordinates of the points on which these calculations are 
founded, must be taken from the data given in $ 2, as here also 
we can use the same planes,of projection as interperpendicular 
planes. (Page 54). For the four otoliths-planes and the six planes 
of the semicircular canals, the magnitudes A, A’, A” etc. can be 
determined; by inserting these values to formula (1), we get the 
size of the angle Q, Q', Q" etc. being what we wished to know. 
(1) 
§ 4. As has been stated the material investigated consisted of 
