198 The Rev. S. Havcuton on the Equilibrium, &c. 
The position and magnitude of the transversals which have been just deter- 
mined, belong to the transversals of a wave whose real vibration takes place in its 
own plane; and the position and magnitude of the transversals belonging to a 
normal vibration might be determined in a manner precisely similar from equa- 
tions (68). 
3rd. Incident vibration normal, and in the plane of incidence. The differen- 
tial equations for this case, and the conditions at the limits, are the equations (63) 
and (65). Representing, therefore, by (9, %, ¢' ¢, ¢”) the phases of the inci- 
dent, reflected normal, refracted normal, reflected transverse, and refracted trans- 
verse waves, the conditions at the limits will become the four following, which 
determine, as before, the four unknown intensities : 
(7 + 7,)siné + 7,,cos0,, = 7’sin@’ + 7’cos0”, 
(7 — 7,)cosd + 7,sin@,, = 7'cos6’ — 7’sin0”, 
1 + 2cos’0 sin26 1-++ 2cos°6" ,, sin26” ts 
Gaps) a, ae = x («! Tica ak yr) (71) 
aanirn ee r x 
sin20 coszo fp esm2o: | eos2e7 
(eae Fa ni = 1 ( rv Ti —) 
The two transversals which I have made use of in this Paper, in investigating 
the conditions at the limits, are merely conceivable mathematical lines, and have 
no real existence in the physical problem: but they are useful to assist in a geo- 
metrical conception of what really happens at the limits. There is one real 
physical vibration, which is to be conceived as accompanied by two mathema- 
tical vibrations, which depend upon the real one—each of the three vibrations 
giving its own conditions at the limits; so that the two transversals, although 
only conceptions of the mind, are naturally suggested by the conditions of the 
problem, and serve to convey a graphic representation of it. 
