MINDING’S THEOREM. 685 
It may do so internally or externally, and consequently the resulting equation 
gives two values of Tz for each value of U7. 
14, This is, in fact, in quaternions identical with the second process em- 
ployed by Professor CurysraL. For, by writing + for p+ Sp in (11) it 
becomes 
7=SpoB , 
and in the same way (13) becomes 
t'—Stat= —SBo’B . 
These, translated into Cartesian scalars, are CHRYSTAL’S equations (8) and 
(9) (Second Method, anté, p. 523). They may be obtained directly by a process 
similar to that in section 8 above. CuHrRysTAL’s first method is, of course, 
included in the solutions afforded by the use of x. 
I may remark, in conclusion, that the process of section 4, leading to an 
equation like (10) above, seems to be the most natural method of applying 
quaternions to questions connected with congruencies. 
