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tokens: scale rad is equal to the radial distance separating adjacent arms near the Sun and the scale azim is the angular distance between them. To reiterate, this method does not afford a reliable estimate of the angle p and the number m but only of rad for a given m . The best-fitting values of , , , , u 0 , v 0 , p and 0 for a given m are found by fitting that minimizes 2 defined as In equation ( 2 ), i counts the N stars in the sample, q is the number of parameters to be derived, v obs, i and v mod, i are the observed and modelled velocities and v obs, i is the uncertainty of v obs, i . The model depends linearly on the set of six parameters (), which can be determined by the standard least-squares fitting for specified 0 , m and p . The quantities and are periodic with respect to 0 with period , while the quantities and are periodic with period 2. For a trailing pattern and the potential , out of the two Suns phase values 0 and 0 that minimize 2 , the values that satisfy the condition are chosen (Mishurov etal. 1997 ; Griv, Ngeow Jiang 2013 , 2014 ; cf. Mishurov etal. 1979 , p. 149). The values of () will completely determine the Galactic differential rotation and the values of ( m , p ) (or rad ) and 0 will determine the nearby spiral pattern. 3 DATA The data of stars were collected from the APOGEE-RC DR13 catalogue of 19291 objects, which is constructed using the method described in Bovy etal. ( 2014 ). This catalogue of intermediate-old giant stars in the near-infrared on scales between 25pc and 10kpc has the estimated uncertainties in the distances and the line-of-sight velocities of 5percent and 0.1kms 1 . Ages of 705 local APOGEE giants with the mean age t age 310 9 yr were estimated by Feuillet etal. ( 2016 ). Details on the APOGEE project in the Northern hemisphere and the dual-hemisphere APOGEE-2 project can be found in Bovy etal. ( 2015 ) and Majewski etal. ( 2016 ), respectively. 1 Fig. 1 shows space locations of all 19291 stars with known d projected on to the Galactic X Y , X Z and Y Z planes. As is seen, the stars are limited mainly within 5kpc from the Sun. In the object distribution, there are no obvious arm-like structures in the X Y plane (Fig. 1 , left-hand panel). A scaleheight of the distribution is h 200300pc (Fig. 1 , middle and right-hand panels). Such a scaleheight h 250pc is typical to intermediate-old population of the local Galactic disc (Kharadze etal. 1989 ; Rix Bovy 2013 ; Camargo, Bica Bonatto 2015 ). Figure 1. The space distribution of all 19296 stars with known photometric distances from the Sun projected on to the Galactic X Y , X Z and Y Z planes. Cartesian X , Y and Z coordinates are shown, the X -axis points to the direction of the Galactic rotation, the Y -axis points to the Galactic anticentre, the Z -axis points to the North Galactic pole, and the axis of the Galactic rotation is taken oriented along the Z -axis, with the Sun at (0, 8.3, 0). The solar circle ( r 0 8.3kpc) and the Z 0 plane are indicated by