1889 - 90 .] Rev. Mr Wilkinson on Theorems in Quaternions. 157 
= C(r, s)(4Y& - 2 Vfc + xSa 4 a 2 - Y£Yx - £Sx) 
+ C (r 4- 1, s — Y)V a^^x 
= C (r, s)Ya 1 a 2 x + C(r + 1, s - l)Ya ± a 2 x ; 
So that Y(2r + 2, 2s + l) = C(r+ 1, s)Ya 1 a^c; . . . (22) 
Again, B(2r +2, 2s 4 1) = B 4 + B 2 + B 3 + B 4 , where, 
B 1 + B 2 = Y. a 1 {S'(2r + 1, 2s + 1) + B'(2 r + 1, 2s + 1) } = 0 ; 
B 1 = Y. ai a 2 {S "(2r, 2s+ 1) + B"(2r, 2s + 1)} 
= C(r, s - l)£Sx ; 
B 3 = C(r, s - ^Ya^Sx ; 
B 4 = Y . Ctga 4 . . . 0'2 r +£i>CL]CL2 a '2r+5' • • • • 
= B (2 r 4* 2, 2s — l)Sa 4 a 2 + Y . a 3 a 4 . . . ft 2 r+ 4 ^^’ a 2 r +5 ••••“••• 
B|(2r + 2, 2s) = Y. £a 3 a 4 . . .a^Sa^. Y. a 3 a 4 . . 02 r+4 S£a 2r+5 . ...... 
B 4 = C(r, s)Y£x - Y.£{S"(2r 4- 1, 2s) 4- B"(2r 4- 1, 2s)} 
= C(r, s) Y£x - C (r - 1, s )£ Sx - C(r, s)Y.£Ys ; 
B x + B 2 + B 3 + B 4 = C(r, s)( Y(x - £Sx - Y. £Yx) = 0 • 
or, B(2r + 2, 2s 4- 1) = 0 ; ..... (23) 
And we can easily show that 
Y(2 r + 2, 2s + 1) - B(2r + 2, 2s + 1) = B(2s + 1, 2r + 2) - Y(2s + 1, 2r + 2) ■ 
so that 
B(2s+ 1, 2r + 2) = Y(2r + 2, 2s + 1) + Y(2s + 1, 2r + 2) 
= C(r+ 1, s)Ya 4 a 2 x; (24) 
So that our formulse (1) to (4) are completely established. 
On an Accidental Illustration of the Effective Ohmic 
Resistance to a Transient Electric Current through 
a Steel Bar. By Sir William Thomson. 
(Read March 17, 1890.) 
After the recent meeting of the British Association at Newcastle, 
Lord Armstrong, in showing me the appliances by which his house 
at Cragside is lighted electrically by water-power, told me of a very 
wonderful incident which he had recently experienced. A bar of 
steel, which he was holding in his hand, was allowed accidentally 
to come in contact with the two poles of a dynamo in action. He 
