348
ON THE STABILITY OF THE SIMPLE SHEARING
[398
is moderately great. For, although capable of evanescence, the functions Ģi> ti, ss> t2 increase in amplitude so rapidly with ?/ that the extreme value of ^ may be said to dominate the integrals. The hyperbolic functions then disappear and the equation reduces* to
*i (%) t* (%) - *2 (%). ti (%) = 0,.....................(40)
TABLE II.
9 * 9 fn 2 I t 2\2 Ŧ!*-*!' Sums of
n Sj2 - if (Sf + ti')* (si2 + *i2)2 fourth column
l + I'OOO 1-000 + 1 -ooo 1-000
3 + 0-997 1-002 + -995 1-995
5 -f 0-951 1-042 + -913 2-908
7 4- 0-581' 1-399 + -415 3-323
9 0-569 2-989 - -191 3-132
1-1 - '3-982 16-60 - -240 2-892
1-3 - 6-155 72-25 - -085 2-807
1-5 + 4-38 485-8 + -009 2-816
1-7 H- 57-9 3660-0 + -016 2-832
1-9 ' + 119-0 31700-0 + -004 2-836
2-1 - 255-0 314000-0 - -001 2-835
2-3 -1854-0 353x10-" - -001 2-834
2-5 - 616-0. 45x10-° - -ooo 2-834
which cannot be satisfied by a moderately large value of y^ For it appears from the appropriate expressions (21)... (24) that the left-hand member of (40) is then
cos(7r/6),
a positive and rapidly increasing quantity. Again; it is evident from Table I that the left-hand member of (32) remains positive for all values of t\z from zero up to some value which must exceed I'l, since up to that point the functions sl} $z, tz are positive while ^ is negative. Even without further examination it seems fairly safe to conclude that (32) cannot be satisfied by any values of ^ and \.
'Another case admitting of simple treatment occurs when 772 and ^ are .both smal], although \ may be great. We have approximately
the next terms being in each case of 6 higher degrees in 77. Thus with omission of terms in rf1 under the integral sign, (31) becomes
f r
- e-^drj.\r
J J
(41)
* Eegard being paid to the character of the functions. Needless to say, it is no general proposition that the value of an integral is determined by the greatest value, however excessive, of the integrand, *.. In neither case can R vanish for a finite (real) value of ?;, and the same is true of Są and 8Z. being blown from a loaded bag, charged beforehand with a foot blower. In this respect they are not fully comparable with those of Prof. Titchener, whose whistle was actuated by squeezing a rubber bulb. However, I have also tried a glass tube, 104 in. long, supported at the middle and rubbed with a resined leather. This should be of the right pitch, but the squeak heard did not suggest an s. I ought perhaps to add that the thing did not work particularly well.