Citing 
Si =S; -—G, 
as expressing the same relation as (7). In this way the difference 
of two vectors is defined. 
We shall speak of the sum and the difference of two vectors not 
only if these relate to the same position of the system, but likewise 
if they are given for different positions. 
§ 6. The material points of the system are said to be connected 
with one another, if the system is, by its nature, only capable of 
such infinitely small displacements as satisfy certain conditions. We 
shall suppose that these may be expressed by ¢ equations of the form 
3n 
Ns tn de, Oldest | RIS), 
1 
in which the coefficients «, are functions of the rectangular coordi- 
nates, but do not explicitly contain the time. Displacements agreeing 
with (9) are called possible displacements ; displacements which violate 
the conditions are however equally imaginable. 
A position of the system and a vector in it being given, we may 
examine if the vector have or not the direction of a possible displa- 
cement. If will have such a direction if its elements or its direction- 
constants obey # equations, similar to (9). 
If two of the three vectors in (7) have the direction of a possible 
displacement, the third will have the same property. 
§ 7. There are directions perpendicular to all possible displace- 
ments. If a vector S is to have such a direction, it must be possible 
to express its elements X, inz quantities 2, by means of the equations 
; 
NN PE Ge gsi ie, (oo at ee ee 
al ao) 
Any system of values for 2, will give a vector that has the 
property in question. 
If, among the vectors occurring in (7), there are two that are 
perpendicular to all possible displacements, the same will be the 
case with the third vector. 
A given vector may be decomposed into two components of which 
the one is perpendicular to all possible displacements, and the other 
has the direction of a possible displacement. There is only one such 
decomposition. 
In order to show this, we may regard as unknown quantities the 
3n elements of the second component and i quantities 5e ud which, 
by (10), the elements of the first component may be expressed. There 
