> as lemmas

1867 W. Thomson & P. G. Tait Treat. Nat. Philos. I. i. 130 A free point has three degrees of freedom, inasmuch as the most general displacement which it can take is resolvable into three, parallel respectively to any three directions, and independent of each other... If the point be constrained to remain always on a given surface, one degree of constraint is introduced, or there are left but two degrees of freedom.

1885 Times 26 Aug. 8/3 The five degrees of freedom being three of translation and two of rotation about two axes.

1936 S. Glasstone Recent Adv. Gen. Chem. ii. 85 This expression gives the whole of the internal entropy, when the vibrational degrees of freedom do not contribute to the total.

1954 H. J. J. Braddick Physics Exper. Method iii. 62 The position of one rigid body relative to another may be defined by six co-ordinates, and it is therefore said to have six degrees of freedom.

1969 Jrnl. Inst. Navigation 22 366 This type of gyro..has a rotor suspended on a spherical air bearing which acts both as a spin bearing and provides for two further degrees of rotational freedom.

1993 R. J. Pond Introd. Engin. Technol. (ed. 2) x. 282 The number of independent ways a robot can move is also known as the robot's degrees of freedom.

2005 Jrnl. Biomechanical Engin. 127 934 The hand mechanism itself has 15 degrees of freedom and five fingers.

1899 R. A. Lehfeldt Text-bk. Physical Chem. v. 208 Such systems may conveniently be called invariant, univariant, divariant, &c., according as they possess no, one, two, &c., degrees of freedom.

1904 A. Findlay Phase Rule ii. 15 We shall therefore define the number of degrees of freedom of a system as the number of the variable factors, temperature, pressure, and concentration of the components, which must be arbitrarily fixed in order that the condition of the system may be perfectly defined.

1948 S. Glasstone Textbk. Physical Chem. (ed. 2) vi. 475 A system consisting of one phase only, e.g., solid, liquid or gaseous, of water has two degrees of freedom,..for..it is necessary to specify both temperature and pressure to define completely the state of the system.

1999 Nature 29 Apr. 755/2 The stochasticity inherent in the climate system's many degrees of freedom.

2001 R. W. Cahn Coming of Materials Sci. iii. 76 Gibbs derived a corollary of general validity, the phase rule , formulated as δ = n + 2 − r. This specifies the number of independent variations δ (usually called ‘degrees of freedom’) in a system of r coexistent phases containing n independent chemical components.

1922 R. A. Fisher in Jrnl. Royal Statist. Soc. 85 88 We shall show that Elderton's Tables of Goodness of Fit..may still be applied, but that the value of n′ with which the table should be entered is not now equal to the number of cells, but to one more than the number of degrees of freedom in the distribution.

1950 G. U. Yule & M. C. Kendall Introd. Theory Statistics (ed. 14) xxi. 485 In the quantity Σ(x − m)² there are n independent contributions of the type (x − m)², and hence we may say that n is the number of degrees of freedom of that estimate; but in the quantity Σ(x − x̄)² we have used the data to estimate x̄, and hence the number of degrees of freedom is lowered by unity, i.e. equals n − 1.

1969 M. G. Kendall & A. Stuart Advanced Theory Statistics (ed. 3) I. xvi. 375 It is thus natural to speak of the number of degrees of freedom, ν, of a function such as χ², meaning thereby that it is distributed as the sum of squares of ν independent standardized normal variates.

1994 P. Ormerod Death of Econ. (1995) 214 (table) Q(3) is the Box–Ljung test statistic of the null hypothesis of white-noise residuals from one through three lags, which has a chi-square distribution with three degrees of freedom.

2004 K. R. Murphy Statist. Power Anal. ii. 37 If the null hypothesis is true and there are 2 and 100 degrees of freedom, then researchers should expect to find F values of 3.09 or lower 95% of the time, and values of 4.82 or lower 99% of the time.